LV 


OUTLINES 

OF 

THEORETICAL  CHEMISTRY 


BY 

FREDERICK  H.   GETMAN,   Ph.D.    (Johns  Hopkins) 

Formerly  Associate  Professor  of  Chemistry  in  Bryn  Mawr  College 


THIRD  EDITION,   THOROUGHLY 
REVISED  AND  ENLARGED 

TOTAL    ISSUE   TEN    THOUSAND 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 
LONDON:  CHAPMAN  &  HALL,  LIMITED 

1922 


COPYRIGHT,  1913,  1918,  1922, 

BY 

FREDERICK   H.   GETMAN 


Stanbopc  [Press 

TECHNICAL  COMPOSITION  COMPANY 

P.  H.  GILSON  COMPANY 

BOSTON,   U.  S.  A. 


To 


WHOSE  TENDER  SYMPATHY  HAS  HELPED  TO 
LIGHTEN  THE  DARKNESS  OF  THE  DAYS 
DURING  WHICH  THESE  PAGES 
WERE  WRITTEN 


49 J 209 


"I  look  upon  the  common  operations  and  practices  of  chymists,  almost 
as  I  do  on  the  letters  of  the  alphabet,  without  whose  knowledge  'tis  very 
hard  for  a  man  to  become  a  philosopher;  and  yet  that  knowledge  is  very  far 
from  being  sufficient  to  make  him  one." 

ROBERT  BOYLE.     (The  Sceptical  Chymist.) 


PREFACE  TO  THIRD  EDITION 

THE  author  has  gladly  availed  himself  of  the  opportunity, 
afforded  by  the  demand  for  a  third  edition  of  his  "  Outlines  of 
Theoretical  Chemistry,"  to  subject  the  entire  text  to  a  thorough 
revision. 

This  undertaking  has  involved  the  rewriting  of  large  portions 
of  the  preceding  edition,  the  introduction  of  much  new  material 
based  upon  recent  advances  in  the  science  of  physical  chemistry, 
and  such  changes  in  the  arrangement  of  the  various  chapters  as 
the  logical  presentation  of  the  subject  demands. 

The  criticisms  and  suggestions  which  have  been  offered  by  a 
number  of  teachers,  who  have  used  the  former  editions  of  the 
book  with  their  classes,  have  proven  most  helpful  in  the  arduous 
task  of  revision. 

The  author  is  especially  indebted  to  Professor  J.  H.  Mathews 
of  the  University  of  Wisconsin,  whose  assistance  has  been  of 
inestimable  value,  not  only  in  connection  with  the  revision  of 
the  text,  but  also  during  the  period  when  the  book  was  passing 
through  the  press.  To  Professor  Farrington  Daniels,  also  of  the 
University  of  Wisconsin,  the  author  is  indebted  for  his  help  in 
reading  the  proof,  while  to  Mr.  Kenneth  Tate,  much  credit  is  due 
for  his  assiduous  care  in  the  preparation  of  the  indices.  To  all 
of  those  friends,  who,  by  their  generous  cooperation,  have  very 
appreciably  lightened  his  labors,  the  author  would  express  his 
sincere  gratitude. 

FREDERICK  H.  GETMAN. 

STAMFORD,  CONN. 
June  20,  1922* 


EXTRACT  FROM  PREFACE  TO  SECOND  EDITION 

THE  demand  for  a  second  edition  of  this  book  has  not  only 
afforded  the  author  an  opportunity  to  thoroughly  revise  the 
original  text,  but  also  has  made  it  possible  to  include  such  new 
material  as  should  properly  find  a  place  in  an  introductory  text- 
book of  theoretical  chemistry. 

The  arduousness  of  the  task  of  revision  and  amplification,  has 
been  appreciably  lightened  by  the  helpful  criticisms  and  valuable 
suggestions,  which  have  been  received  from  those  who  have  used 
the  first  edition  with  their  classes. 

So  numerous  were  the  additional  topics  suggested  that  the 
author  found  himself  confronted  with  a  veritable  embarrassment 
of  riches,  and  not  the  least  difficult  part  of  his  task  has  been  the 
attempt  to  weave  in  as  many  of  these  suggestions  as  seemed  to  be 
consistent  with  a  well-balanced  presentation  of  the  entire  subject. 

The  features  which  distinguish  this  edition  from  the  preceding 
edition  may  be  briefly  summarized  as  follows:  — 

1.  The  necessity  of  introducing  a  short  chapter  on  the  mod- 
ern conception  of  the  atom  and  its  structure,  involved  the 
further  necessity  of  including  a  preliminary  chapter,  treat- 
ing of  those  radioactive  phenomena  upon  which  the  greater 
part  of  our  present  atomic  theory  is  based. 

2.  The   chapter   on   solids   has   been   practically  rewritten: 
the  space  formerly  devoted  to  an  outline  of  crystallog- 
raphy, being  devoted,  in  the  present  edition,  to  a  discussion 
of  the  absorption  of  heat  by  crystalline  solids,  and  the  bear- 
ing of  X-ray  spectra  on  crystalline  form. 

3.  The  increasing  importance  of  colloidal  phenomena,  not 
only  to  the  chemist,  but  also  to  the  biologist,  to  the  physi- 
cian, and  to  the  technologist,  has  made  it  seem  desirable 
to  rewrite  the  entire  chapter  devoted  to  the  chemistry  of 
colloids. 

4.  The  Brownian  movement,    and    its    bearing    upon    the 
existence  of  molecules,   has  been  briefly  presented  in  a 
separate  chapter,  in  order  to  emphasize  the  importance  of 

vii 


viii  PREFACE  TO  SECOND  EDITION 

the  brilliant  experimental  work  of  Perrin,  and  others,  in 
confirming  the  kinetic  theory. 

5.  The   chapter  treating   of  electromotive  force  has   been 
enlarged,  so  as  to  include  a  discussion  of  some  of  the  more 
valuable  methods  which  have  been  proposed  for  deter- 
mining junction  potentials,  and  also  to  point  out,  several 
useful  applications  of  concentration  cells. 

6.  An  entirely  new  chapter,  in  which  an  attempt  has  been 
made  to  present  the  salient  facts  and  more  important 
theories  of  photochemistry  in  succinct  form,  replaces  the 
former  chapter  treating  of  the  relations  between  radiant 
and  chemical  energy. 

FREDERICK  H.  GETMAN. 

STAMFORD,  CONN. 
Aug.  7,  1918. 


EXTRACT  FROM  PREFACE  TO  FIRST  EDITION 

"The  last  thing  that  we  find  in  making  a  book  is  to  know  what  we  must 
put  first."  —  PASCAL. 

THE  present  book  is  designed  to  meet  the  requirements  of 
classes  beginning  the  study  of  theoretical,  or  physical  chemistry. 
A  working  knowledge  of  elementary  chemistry  and  physics  has 
been  presupposed  in  the  presentation  of  the  subject,  the  introduc- 
tory chapter  being  the  only  portion  of  the  book  in  which  space 
is  devoted  to  a  review  of  principles  with  which  the  student  is 
assumed  to  be  already  familiar.  With  the  exception  of  a  few  para- 
graphs in  which  the  application  of  the  calculus  is  unavoidable, 
no  use  is  made  of  the  higher  mathematics,  so  that  the  book  should 
be  intelligible  to  the  student  of  very  moderate  mathematical 
attainments.  Wherever  the  calculus  has  been  employed,  the 
student  who  is  unfamiliar  with  this  useful  tool  must  accept  the 
correctness  of  the  results,  without  attempting  to  follow  the  suc- 
cessive operations  by  which  they  are  obtained. 

The  contributions  to  our  knowledge  in  the  domain  of  physical 
chemistry  have  increased  with  such  rapidity  within  recent  years, 
that  the  prospective  author  of  a  general  text-book  finds  himself 
confronted  with  the  vexing  problem,  of  what  to  omit  rather  than 
what  to  include.  In  selecting  material  for  this  book,  the  author 
has  been  guided,  in  large  measure,  by  his  own  experience  in  teach- 
ing theoretical  chemistry  to  beginners,  and  to  advanced  students. 
The  attempt  has  been  made  to  present  the  more  difficult  portions 
of  the  subject,  such  as  the  osmotic  theory  of  solutions,  the  laws 
of  equilibrium  and  chemical  action,  and  the  principles  of  electro- 
chemistry, in  a  clear  and  logical  manner.  While  the  treatment 
of  each  topic  is  necessarily  brief,  yet  the  effort  has  been  made  to 
avoid  the  sacrifice  of  clearness  to  brevity. 

The  author  is  fully  convinced  from  his  own  experience,  as  well 
as  from  that  of  his  colleagues,  that  the  complete  mastery  of  the 
fundamental  principles  of  the  science,  is  best  attained  through 
the  solution  of  numerical  examples.  For  this  reason,  typical 

ix 


X  PREFACE  TO  FIRST  EDITION 

problems    have    been    appended    to    various    chapters    of    the 
book. 

Numerous  references  to  original  papers  have  been  given  through- 
out, since  the  importance  of  literary  research,  on  the  part  of  the 
student,  is  conceded  by  all  teachers  to  .be  of  prime  importance. 

FREDERICK  H.   GETMAN. 

STOCKBRIDGE.  MASS. 
Aug.  18,  1913. 


TABLE   OF   CONTENTS 

CHAP.  PAGE 

PREFACE  TO  THIRD  EDITION v 

EXTRACT  FROM  PREFACE  TO  SECOND  EDITION vn 

EXTRACT  FROM  PREFACE  TO  FIRST  EDITION rx 

I.   FUNDAMENTAL  PRINCIPLES 1 

II.   GASES 21 

III.  LIQUIDS 53 

IV.  SOLIDS 85 

V.   RELATION  BETWEEN  PHYSICAL  PROPERTIES  AND  MOLECULAR 

CONSTITUTION 101 

VI.   ELEMENTARY  PRINCIPLES  OF  THERMODYNAMICS 130 

VII.   SOLUTIONS 149 

VIII.   DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE 174 

IX.   SOLUTIONS  OF  ELECTROLYTES 210 

X.   COLLOIDS 226 

XI.    THERMOCHEMISTRY 273 

XII.   HOMOGENEOUS  EQUILIBRIUM 302 

XIII.  HETEROGENEOUS  EQUILIBRIUM 322 

XIV.  CHEMICAL  KINETICS .'  360 

XV.   ELECTRICAL  CONDUCTANCE 389 

XVI.   ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS 430 

XVII.   ELECTROMOTIVE  FORCE 457 

XVIII.   ELECTROLYSIS  AND  POLARIZATION 511 

XIX.   PHOTOCHEMISTRY 524 

XX.  CLASSIFICATION  OF  THE  ELEMENTS 546 

XXI.    ELECTRICAL  THEORY  OF  MATTER 556 

XXII.   RADIOACTIVITY 569 

XXIII.  ATOMIC  STRUCTURE 586 

Index  of  Authors 611 

Index  of  Subjects 617 

xi 


THEORETICAL  CHEMISTRY 

CHAPTER  I 
FUNDAMENTAL  PRINCIPLES 

Theoretical  Chemistry.  That  portion  of  the  science  of  chem- 
istry which  has  for  its  object  the  study  of  the  laws  controlling 
chemical  phenomena  is  called  theoretical  or  physical  chemistry. 
The  first  attempt  to  summarize  the  more  important  facts  and 
ideas  underlying  the  science  of  chemistry  was  made  by  Dalton  in 
1808  in  his  "  New  System  of  Chemical  Philosophy."  The  birth 
of  the  science  of  theoretical  chemistry  may  be  considered  to  be 
coeval  with  the  appearance  of  Dalton's  epoch-making  book. 

Theoretical  chemistry  is  concerned  with  the  great  generaliza- 
tions of  chemical  science  and  bears  the  same  relation  to  chemistry 
that  philosophy  bears  to  the  whole  body  of  scientific  truth;  it 
aims  to  systematize  all  of  the  established  facts  of  chemistry  and 
to  discover  the  laws  governing  the  various  phenomena  of  chemical 
action.  ^ 

Law,  Hypothesis  and  Theory.  The  science  of  chemistry  is 
based  upon  experimentally  .established  facts.  When  a  number 
of  facts  have  been  collected  and  classified  we  may  proceed  to  draw 
inferences  as  to  the  behavior  of  systems  under  conditions  which 
have  not  been  investigated.  This  process  of  reasoning  by  analogy 
we  term  generalization  and  the  conclusion  reached  we  call  a  law. 
It  is  apparent  that  a  law  is  not  an  expression  of  an  infallible  truth, 
but  it  is  rather  a  condensed  statement  of  facts  which  have  been 
discovered  by  experiment.  It  enables  us  to  predict  results  with- 
out recourse  to  experiment.  The  fewer  the  number  of  cases  in 
which  a  law  has  been  found  to  be  invalid,  the  greater  becomes  our 
confidence  in  it,  until  eventually  it  may  come  to  be  regarded  as 
tantamount  to  a  statement  of  fact. 

Natural  laws  may  be  discovered  either  by  the  correlation  of  ex- 
perimentally determined  facts,  as  outlined  above,  or  by  means  of  a 

1 


2  •        • '  •      ;  .THEORETICAL  CHEMISTRY 

speculation  as  to  the  probable  cause  of  the  phenomenon  in  question. 
Such  ,a  speculation  in  regard  to  the  cause  of  a  phenomenon  is 
called  an  hypothesis. 

After  an  hypothesis  has  been  subjected  to  the  test  of  experiment 
and  has  been  shown  to  apply  to  a  large  number  of  closely  related 
phenomena  it  is  termed  a  theory. 

In  his  address  to  the  British  Association  (Dundee,  1912),  Profes- 
sor Senier  has  this  to  say:  "  While  the  method  of  discovery  in 
chemistry  may  be  described  generally,  as  inductive,  still  all  the 
modes  of  inference  which  have  come  down  to  us  from  Aristotle, 
analogical,  inductive,  and  deductive,  are  freely  used.  An  hy- 
pothesis is  framed  which  is  then  tested,  directly  or  indirectly, 
by  observation  and  experiment.  All  the  skill,  and  all  the  resource 
the  inquirer  can  command,  are  brought  into  his  service;  his  work 
must  be  accurate;  and  with  unqualified  devotion  to  truth  he 
abides  by  the  result,  and  the  hypothesis  is  established,  and  be- 
comes a  part  of  the  theory  of  science,  or  is  rejected  or  modified." 

Elements  and  Compounds.  All  definite  chemical  substances 
are  divided  into  two  classes,  elements  and  compounds. 

Robert  Boyle  was  the  first  to  make  this  distinction.  He  de- 
fined an  element  as  a  substance  which  is  incapable  of  resolution 
into  anything  simpler.  The  substances  formed  by  the  chemical 
combination  of  two  or  more  elements  he  termed  chemical  com- 
pounds. This  definition  of  an  element  as  given  by  Boyle  was 
later  proposed  by  Lavoisier  and,  notwithstanding  the  vast  accumu- 
lation of  scientific  knowledge  since  their  time,  the  definition  re- 
mains very  satisfactory  today. 

At  the  present  time  we.  have  a  group  of  about  ninety  substances 
which  have  resisted  all  efforts  to  decompose  them  into  simpler 
substances.  These  are  the  so-called  chemical  elements.  It 
should  be  borne  in  mind,  however,  that  because  we  have  failed  to 
resolve  these  substances  into  simpler  forms  of  matter,  we  are  not 
warranted  in  maintaining  that  such  resolution  may  not  be  effected 
in  the  future. 

Recent  investigations  of  the  radioactive  elements  have  shown 
that  they  are  continuously  undergoing  a  series  of  transformations, 
one  of  the  products  of  which  is  the  inactive  element  helium.  This 
behavior  is  contrary  to  the  old  view  that  transformation  of  one 
element  into  another  is  impossible.  At  first  the  attempt  was 
made  to  explain  it  by  assuming  that  the  radioactive  element  was 


FUNDAMENTAL  PRINCIPLES  3 

a  compound  of  helium  with  another  element,  but  since  the  radio- 
active elements  possess  all  of  the  properties  characteristic  of 
elements  as  distinguished  from  compounds,  and  find  appropriate 
places  in  the  periodic  table  of  Mendeleeff,  the  "  compound  theory  " 
must  be  abandoned.  Uranium  and  thorium,  the  heaviest  ele- 
ments known,  appear  to  be  undergoing  a  process  of  spontaneous 
disintegration  over  which  we  have  no  control.  The  products 
of  this  disintegration  have  filled  the  gap  in  the  periodic  table 
between  thorium  and  lead  with  about  thirty  new  elements,  each 
of  which  is  in  turn  undergoing  transformations  similar  to  those 
of  the  parent  elements.  Professor  Soddy*  says:  "  In  spite  of 
the  existence  at  one  time  of  a  vague  belief  (a  belief  which  has  no 
foundation),  that  all  matter  maybe  to  a  certain  extent  radioactive, 
just  as  all  matter  is  believed  to  be  to  a  certain  extent  magnetic,  it 
is  recognized  today  that  radioactivity  is  an  exceedingly  rare  prop- 
erty of  matter." 

Notwithstanding  these  remarkable  discoveries,  we  may  still 
hold  to  the  idea  of  an  element  as  suggested  by  Boyle  and  Lavoisier. 
Professor  Walker  says :  "  The  elements  form  a  group  of  substances, 
singular  not  only  with  respect  to  the  resistance  which  they  offer 
to  decomposition,  but  also  with  respect  to  certain  regularities 
displayed  by  them  and  not  shared  by  substances  which  are  desig- 
nated as  compounds." 

Law  of  the  Conservation  of  Mass.  In  1774,  as  the  result  of 
a  series  of  experiments,  Lavoisier  established  the  law  of  the  con- 
servation of  mass  which  may  be  stated  as  follows:  In  a  chemical 
reaction  the  total  mass  of  the  reacting  substances  is  equal  to  the  total 
mass  of  the  products  of  the  reaction.  It  is  sometimes  stated  thus:  — 
the  total  mass  of  the  universe  is  a  constant;  but  this  form  of 
statement  is  open  to  the  objection  that  we  have  no  means  of  verifi- 
cation; it  is  a  statement  of  a  fact  which  transcends  our  expe- 
rience. 

The  law  of  the  conservation  of  mass  has  been  subjected  to  most 
rigid  investigation  by  Landoltf  in  a  series  of  experiments  extend- 
ing over  a  period  of  fifteen  years. 

The  reacting  substances,  A  B  and  CD,  were  placed  in  the  two 
arms  of  the  inverted  U-tube  shown  in  Fig.  1  which  was  then  sealed 
at  S  and  the  whole  weighed  upon  an  extremely  sensitive  balance. 

*  Chemistry  of  the  Radio-Elements,  p.  2. 

f  Zeit.  phys.  chem.,  12,  1  (1893);  55,  589  (1906). 


THEORETICAL  CHEMISTRY 


S 
A 


The  vessel  was  then  inverted  when  the  following  reaction  took 
place  :  — 

AB  +  CD  -»  AD  +  CB. 

When  the  reaction  was  complete  and  sufficient  time  had  elapsed 
to  allow  the  vessel  to  return  to  its  original  volume  (this  some- 
times required  nearly  three  weeks),  it  was 
weighed  again,  using  every  precaution  to 
avoid  errors,  and  any  gain  or  loss  in  weight 
noted. 

Landolt  concluded,  from  the  thirty  or 
more  reactions  which  he  studied,  that  the 
gain  or  loss  in  weight  was  less  than  one  ten- 
millionth  of  the  total  weight.* 

Law  of  Definite  Proportions.  The  enun- 
ciation of  the  law  of  the  conservation  of 
mass  and  the  introduction  of  the  balance 
into  the  chemical  laboratory  marked  the 
beginning  of  a  new  era  in  the  history  of 
chemistry,  —  the  era  of  quantitative  chem- 
istry. As  the  result  of  painstaking  experi- 


AB 


Fig.   1 


mental  work,  Richter  and  Proust  announced  the  law  of  definite 
proportions  about  the  beginning  of  the  nineteenth  century.  This 
law  may  be  expressed  thus:  A  definite  chemical  compound  always 
contains  the  same  elements  united  in  the  same  proportion  by  weight. 
Shortly  after  the  enunciation  of  this  law  its  truth  was  ques- 
tioned by  the  French  chemist  Berthollet.f  From  the  results  of  a 
series  of  brilliant  experiments,  he  became  convinced  that  chemical 
reactions  are  largely  controlled  by  the  relative  amounts  of  the 
reacting  substances."  As  v/e  shall  see  later,  he  really  foreshadowed 
the  work  of  Guldberg  and  Waage  who  were  the  first  to  correctly 
formulate  the  influence  of  mass  on  a  chemical  reaction.  Ber- 
thollet  argued  that  when  two  elements  unite  to  form  a  compound, 
the  proportion  of  one  of  the  elements  in  the  compound  is  condi- 
tioned solely  by  the  amount  of  that  element  which  is  available. 
This  led  to  the  celebrated  controversy  between  Berthollet  and 

*  An  excellent  summary  of  this  important  investigation  will  be  found  in 
the  Journal  de  Chimie  physique,  6,  625  (1908).  For  later  investigations 
on  the  conservation  of  mass,  see  paper  by  Manley,  Phil.  Trans.  Roy.  Soc. 
A  212,  227  (1913). 

t  Essai  de  statique  chimique  (1803). 


FUNDAMENTAL  PRINCIPLES  5 

Proust  which  finally  resulted  in  the  establishment  of  the 
latter's  original  statement.  Subsequent  investigation  has  only 
strengthened  our  faith  in  the  law  of  definite  proportions. 

Law  of  Multiple  Proportions.  Elements  are  known  to  unite 
in  more  than  one  proportion  by  weight.  Dalton  analyzed  the 
two  compounds  of  carbon  and  hydrogen,  methane  and  ethylene, 
and  found  that  the  ratio  of  the  weights  of  carbon  to  hydrogen  in 
the  former  was  6  :  2  while  in  the  latter  it  was  6:1.  That  is,  for 
the  same  weight  of  carbon,  the  weights  of  the  hydrogen  in  the  two 
compounds  were  in  the  ratio  2:1. 

A  large  number  of  compounds  were  examined  and  similar 
simple  ratios  between  the  masses  of  the  constituent  elements 
were  found.  As  a  result  of  these  observations,  Dalton*  formu- 
lated in  1808  the  law  of  multiple  proportions,  as  follows:  When 
two  elements  unite  in  more  than  one  proportion,  for  a  fixed  mass  of 
one  element  the  masses  of  the  other  element  bear  to  each  other  a  simple 
ratio.  Notwithstanding  the  fact  that  Dalton  was  a  careless 
experimenter,  the  subsequent  investigations  of  Marignac  and 
others  have  established  the  validity  of  his  law. 

Law  of  Combining  Proportions.  Dalton  pointed  out  that  it  is 
possible  to  assign  to  every  element  a  definite  relative  weight  with 
which  it  enters  into  chemical  combination.  He  observed  that 
the  weights,  or  simple  multiples  of  the  weights,  of  the  different 
elements  which  unite  with  a  given  weight  of  a  definite  element, 
represent  the  weights  of  the  different  elements  which  combine 
with  each  other.  The  weights  of  the  elements  which  combine 
with  each  other  are  termed  their  combining  weights.  This  com- 
prehensive law  of  chemical  combination  may  be  stated  as  follows: 
Elements  combine  in  the  ratio  vf  tfyeir  combining  weights^  or 
in  simple  multiples  of  this  ratio.  It  will  be  observed  that  this  law 
really  includes  the  law  of  definite  and  the  law  of  multiple  pro- 
portions. 

If  we  assume  the  combining  weight  of  hydrogen  to  be  unity, 
the  combining  weights  of  chlorine,  oxygen  and  sulphur  will  be 
35.5,  8  and  16  respectively.  These  numbers  represent  the  ratios 
in  which  the  elements  substitute  each  other  in  chemical  com- 
pounds. Hydrochloric  acid,  for  example,  contains  35.5  parts  by 
weight  of  chlorine  to  1  part  by  weight  of  hydrogen  and  when 
oxygen  is  substituted  for  chlorine,  forming  water,  the  new  com- 
*  A  New  System  of  Chemical  Philosophy  (1808). 


6  THEORETICAL  CHEMISTRY 

pound  contains  8  parts  by  weight  of  oxygen  to  1  part  by  weight 
of  hydrogen.  Similarly,  if  the  oxygen  be  substituted  by  sulphur, 
forming  hydrogen  sulphide,  there  will  be  found  16  parts  by  weight 
of  sulphur  to  1  part  by  weight  of  hydrogen.  We  may  say,  then, 
that  35.5  parts  of  chlorine,  8  parts  of  oxygen  and  16  parts  of  sul- 
phur are  equivalent. 

A  chemical  equivalent  may  be  defined  as  the  weight  of  an  element 
which  is  necessary  to  combine  with,  or  displace,  1  part  by  weight 
of  hydrogen. 

The  Atomic  Theory.  In  very  early  times  two  different  views 
were  entertained  by  opposing  schools  of  Greek  philosophers  as  to 
the  mechanical  constitution  of  matter.  According  to  the  school 
of  Plato  and  Aristotle,  matter  was  thought  to  be  continuous 
within  the  space  it  appears  to  fill  and  to  be  capable  of  indefinite 
subdivision.  According  to  the  other  school,  first  taught  by  Leu- 
cippus,  and  afterwards  by  Democritus  and  Epicurus,  matter 
was  considered  to  be  made  up  of  primordial,  extremely  minute 
particles,  distinct  and  separable  from  each  other  but  in  themselves 
incapable  of  division.  These  ultimate  particles  were  called  atoms 
(^aroAtos),  signifying  something  indivisible.  While  the  Aristote- 
lian doctrine  held  sway  for  many  centuries  yet  the  notion  of  atoms 
was  revived  at  intervals.  Late  in  the  seventeenth  century,  Boyle 
seems  to  have  looked  upon  chemical  combination  as  the  result  of 
atomic  association. 

Guided  by  these  early  speculations  as  to  the  constitution  of 
matter  and  influenced  by  his  study  of  the  writings  of  Sir  Isaac 
Newton,  Dalton  seems  to  have  formed  a  mental  picture  of  the 
part  played  by  atoms  in  the  act  of  chemical  combination.  After 
a  few  carelessly  performed  experiments,  the  results  of  which  ac- 
corded with  his  preconceived  ideas,  he  formulated  his  atomic 
theory. 

According  to  this  theory  matter  is  composed  of  extremely 
minute,  indivisible  particles  or  atoms.  Atoms  of  the  same  ele- 
ment are  all  of  equal  weight,  but  atoms  of  different  elements  have 
weights  proportional  to  their  combining  numbers.  Chemical 
compounds  are  formed  by  the  union  of  atoms  of  different  kinds. 
This  theory  offers  a  simple,  rational  explanation  of  the  laws  of 
chemical  combination. 

Since  a  chemical  compound  results  from  the  union  of  atoms, 
each  of  which  has  a  definite  weight,  its  composition  must  be  in- 


FUNDAMENTAL  PRINCIPLES  7 

variable,  —  which  is  the  law  of  definite  proportions.  Again, 
when  atoms  combine  in  more  than  one  proportion,  for  a  fixed 
weight  of  atoms  of  one  kind,  the  weights  of  the  other  species  of 
atoms  must  bear  to  each  other  a  simple  ratio,  since  the  atoms  are 
indivisible  units.  This  is  clearly  the  law  of  multiple  propor- 
tions. 

Finally,  the  law  of  combining  weights  is  seen  to  follow  as  a 
necessary  consequence  of  the  atomic  theory,  since  the  experimen- 
tally determined  combining  weights  bear  a  simple  relation  to  the 
relative  weights  of  the  atoms. 

At  the  time  when  Dalton  proposed  his  atomic  theory,  the  num- 
ber of  facts  to  be  explained  was  comparatively  small,  but  with 
the  enormous  growth  of  the  science  of  chemistry  during  the  past 
century  and  with  the  vast  accumulation  of  data,  the  theory  has 
proved  capable  of  affording  adequate  representation  of  all  of 
the  facts,  and  has  opened  the  way  to  many  important  generaliza- 
tions. 

While  the  atomic  theory  has  played  a  very  important  part  in  the 
development  of  modern  chemistry,  and  while  we  recognize  that  it 
helps  to  clarify  our  thinking  and  enables  us  to  construct  a  mental 
image  of  tiny  spheres  uniting  to  form  a  chemical  compound,  yet  we 
must  not  forget  the  fact  that  these  atoms  are  purely  hypothetical. 

Faraday  has  said:  "Whether  matter  be  atomic  or  not,  this 
much  is  certain,  that  granting  it  to  be  atomic,  it  would  appear 
as  it  now  does." 

Combining  Weights  and  Atomic  Weights.  The  problem  of 
determining  the  relative  atomic  weights  of  the  elements  would  at 
first  sight  appear  to  be  a  very  simple  matter.  This  might  appar- 
ently be  accomplished  by  selecting  one  element,  say  hydrogen, 
it  being  the  lightest  known  element,  as  the  standard;  a  compound 
of  hydrogen  and  another  element  may  then  be  analyzed  and  the 
amount  of  the  other  element  in  combination  with  one  part  by 
weight  of  hydrogen  determined.  This  weight  will  be  its  atomic 
weight  only  when  the  compound  contains  but  one  atom  of  each 
element.  To  determine  the  relative  atomic  weight,  therefore,  we 
must  know,  in  addition  to  the  chemical  equivalent  of  the  element, 
the  number  of  atoms  present  in  the  compound.  For  example,  the 
analysis  of  water  shows  it  to  contain  8  parts  by  weight  of  oxygen 
to  1  part  by  weight  of  hydrogen;  the  chemical  equivalent  of 
oxygen  is,  therefore,  8,  and  if  water  contained  but  one  atom  of 


8  THEORETICAL  CHEMISTRY 

hydrogen  the  atomic  weight  of  oxygen  would  be  8.  It  can  be 
shown,  however,  that  water  contains  two  atoms  of  hydrogen 
and  one  atom  of  oxygen,  therefore,  the  atomic  weight  of  oxygen 
must  be  16.  It  is  evident,  therefore,  that  neither  the  analysis 
nor  the  synthesis  of  a  compound  is  sufficient  to  enable  us  to  deter- 
mine the  number  of  atoms  of  an  element  combined  with  one  atom 
of  hydrogen.  We  shall  proceed  to  the  consideration  of  the 
methods  by  which  this  problem  may  be  solved. 

Gay-Lussac's  Law  of  Volumes.  Gay-Lussac,  in  1808,  while 
studying  the  densities  of  gases  before  and  after  reaction, 
announced  the  following  law:  When  gases  combine  they  do  so  in 
simple  ratios  by  volume,  and  the  volume  of  the  gaseous  product  bears 
a  simple  ratio  to  the  volumes  of  the  reacting  gases  when  measured 
under  like  conditions  of  temperature  and  pressure.  Thus,  one  vol- 
ume of  hydrogen  combines  with  one  volume  of  chlorine  to  form 
two  volumes  of  hydrochloric  acid ;  one  volume  of  oxygen  combines 
with  two  volumes  of  hydrogen  to  form  two  volumes  of  water  (va- 
por) ;  and  one  volume  of  nitrogen  combines  with  three  volumes  of 
hydrogen  to  form  two  volumes  of  ammonia. 

In  a  previous  investigation,  Gay-Lussac  had  shown  that  all 
gases  behave  identically  when  subjected  to  changes  of  temper- 
ature and  pressure.  This  fact,  taken  together  with  the  simple 
volumetric  relation  just  enunciated  and  the  atomic  theory,  sug- 
gested a  possible  relation  between  the  number  of  ultimate  particles 
in  equal  volumes  of  different  gases. 

Berzelius  attempted  to  show  that  under  corresponding  condi- 
tions of  temperature  and  pressure,  equal  volumes  of  different 
gases  contain  the  same  number  of  atoms,  but  he  was  compelled 
to  abandon  the  assumption  as  untenable. 

Avogadro's  Hypothesis.  It  remained  for  the  Italian  physicist, 
Avogadro,*  in  1811,  to  point  out  the  distinction  between  atoms 
and  molecules,  terms  which  had  been  used  almost  synonymously 
up  to  his  time.  He  defined  the  atom  as  the  smallest  particle 
which  can  enter  into  chemical  combination,  whereas  the  molecule 
is  the  smallest  portion  of  matter  which  can  exist  in  a  free  state. 
He  then  formulated  the  following  hypothesis  :f  Under  the  same 
conditions  of  temperature  and  pressure,  equal  volumes  of  all  gases 
contain  the  same  number  of  molecules.  This  hypothesis  has  been 

*  Jour.  Phys.  73,  58  (1811). 

t  Ampere  advanced  practically  the  same  hypothesis  in  1814. 


FUNDAMENTAL  PRINCIPLES  9 

subjected  to  such  rigid  experimental  and  mathematical  tests  that 
its  validity  cannot  be  questioned. 

Avogadro's  Hypothesis  and  Molecular  Weights.  According 
to  Gay-Lussac,  when  hydrogen  and  oxygen  unite  to  form  water 
vapor,  two  volumes  of  hydrogen  combine  with  one  volume  of 
oxygen  yielding  two  volumes  of  water  vapor.  According  to  the 
hypothesis  of  Avogadro,  the  number  of  molecules  of  water  vapor 
is  double  the  number  of  molecules  of  oxygen,  and,  since  each  mole- 
cule of  water  vapor  must  contain  at  least  one  atom  of  oxygen, 
it  follows  that  the  molecule  of  oxygen  must  contain  at  least  two 
atoms  of  oxygen.  Since  the  value  of  the  atomic  weight  of  oxy- 
gen is  16,  it  is  evident  that  its  molecular  weight  must  be  32. 

It  is  convenient  to  express  molecular  and  atomic  weights  in 
terms  of  the  same  unit,  for  then  the  molecular  weight  of  a  sub- 
stance will  be  simply  the  sum  of  the  weights  of  the  atoms  con- 
tained in  the  molecule.  The  determination  of  the  approximate 
molecular  weight  of  a  substance,  therefore,  resolves  itself  into 
ascertaining  the  mass  of  its  vapor  in  grams  which,  under  the 
same  conditions  of  temperature  and  pressure,  will  occupy  the 
same  volume  as  32  grams  of  oxygen. 

This  weight  is  called  the  gram-molecular  weight  or  the  molar 
weight  of  the  substance,  while  the  corresponding  volume  is  known 
as  the  gram-molecular  or  molar  volume.  It  is  nearly  the  same  for 
all  gases,  and  at  0°  and  760  mm.  it  may  be  taken  equal  to  22.4 
liters.  The  molecular  weights  obtained  from  vapor  density  meas- 
urements are  approximate  only,  because  of  the  failure  of  most 
gases  and  vapors  to  obey  the  simple  gas  laws,  a  condition  essen- 
tial to  the  strict  applicability  of  Avogadro's  hypothesis 

Atomic  Weights  from  Molecular  Weights.  While  vapor  den- 
sity determinations  as  ordinarily  carried  out  do  not  give  exact 
molecular  weights,  it  is  an  easy  matter  to  arrive  at  the  true  values 
when  we  take  into  consideration  the  results  of  chemical  analysis. 
It  is  apparent  that  the  true  molecular  weight  must  be  the  sum  of 
the  weights  of  the  constituent  elements,  these  weights  being  exact 
multiples  or  submultiples  of  their  combining  proportions,  which 
proportions  have  been  determined  by  analysis  alone.  We  select, 
as  the  true  molecular  weight,  the  value  which  is  nearest  to  the 
approximate  molecular  weight  calculated  from  the  vapor  density 
of  the  substance.  For  example,  the  molecular  weight  of  ammonia, 
as  computed  from  its  vapor  density,  is  17.5  or,  in  other  words, 


10 


THEORETICAL  CHEMISTRY 


17.5  grams  of  ammonia  occupy  the  same  volume  as  32  grams  of 
oxygen,  measured  under  the  same  conditions  of  temperature 
and  pressure.  The  analysis  of  ammonia  shows  us  that  for  every 
gram  of  hydrogen,  there  are  present  4.67  grams  of  nitrogen. 
Hence  the  true  molecular  weight  must  contain  a  multiple  of  1 
gram  of  hydrogen  and  the  same  multiple  of  4.67  grams  of  nitro- 
gen. The  problem  is,  to  find  what  integral  value  must  be  as- 
signed to  x  in  the  expression,  x  (1  +  4.67),  in  order  that  it  may 
give  the  closest  approximation  to  17.5.  Clearly  if  x  =  3  the 
value  of  the  expression  becomes  17,  and  this  we  take  to  be  the 
true  molecular  weight.  This  gives  3  X  4.67  —  14  as  the  probable 
atomic  weight  of  nitrogen.  To  decide  whether  the  atomic  weight 
of  nitrogen  is  a  multiple  or  a  submultiple  of  14,  we  must  deter- 
mine the  molecular  weights  of  a  large  number  of  gaseous  or  vapor- 
izable  compounds  of  nitrogen,  and  select  as  the  atomic  weight 
the  smallest  quantity  of  the  element  which  is  present  in  any  one 
of  them. 

The  following  table  gives  a  list  of  seven  gaseous  compounds  of 
nitrogen  together  with  their  gram-molecular  weights,  and  the 
number  of  grams  of  the  element  in  the  gram-molecule. 


Compound 

Gram-mol. 
Wt. 

Grams  Nitro- 
gen. 

Ammonia  

17 

14 

Nitric  oxide  

30 

14 

Nitrogen  peroxide 

46 

14 

Methyl  nitrate 

77 

14 

Cyanogen  chloride       

61  5 

14 

Nitrous  oxide  

44 

28 

52 

28 

It  will  be  observed  that  the  least  weight  of  nitrogen  entering 
into  a  gram-molecular  weight  of  any  of  these  compounds  is  14 
grams,  and,  therefore,  we  accept  this  value  as  the  atomic  weight 
of  the  element,  although  there  is  still  a  very  slight  chance  that 
in  some  other  compound  of  nitrogen  a  smaller  weight  of  the  ele- 
ment may  be  found.  We  shall  now  proceed  to  point  out  that 
there  are  methods  by  which  the  probable  values  of  the  atomic 
weights  may  be  checked. 

Specific  Heat  and  Atomic  Weight.  In  1819,  the  French  chem- 
ists, Dulong  and  Petit,*  pointed  out  a  very  simple  relation  be- 
*  Ann.  Chim.  Phys.,  10,  395  (1819). 


FUNDAMENTAL  PRINCIPLES 


11 


tween  the  specific  heats  of  the  elements  in  the  solid  state  and  their 
atomic  weights.  This  relation,  known  as  the  law  of  Dulong  and 
Petit,  is  as  follows :  The  product  of  the  specific  heat  and  the  atomic 
weight  of  the  solid  elements  is  constant.  The  value  of  this  constant, 
called  the  atomic  heat,  is  approximately  6.4.  A  little  reflection 
will  show  that  an  alternative  statement  of  this  law  is  that  the 
atoms  of  the  elements  in  the  solid  state  have  the  same  thermal  capac- 
ity. The  specific  heats,  atomic  weights  and  atomic  heats  of 
several  elements  are  given  in  the  subjoined  table. 

ATOMIC  HEATS 


Element. 

At.  Wt. 

Sp.  Ht. 

At.  Ht. 

Lithium.                                

7 

0  940 

6  6 

Glucinum                           

9 

0.410 

3  7 

Boron  (amorphous)  

11 

0.250 

2  8 

Carbon  (diamond)  

12 

0.140 

1  7 

Sodium                        

23 

0.290 

6  7 

Silicon  (crystalline)  
Potassium       

28 
39 

0.160 
0.166 

4.5 
6  5 

Calcium         

40 

0.170 

6.8 

Iron 

56 

0  112 

6  3 

Copper 

63 

0  093 

5  9 

Zinc                                           

65 

0  093 

6  1 

Silver                          

108 

0  056 

6  0 

Tin                       

119 

0.054 

6  5 

Gold                 

197 

0.032 

6.3 

Mercury  

200 

0.032 

6.4 

It  is  truly  remarkable  that  elements  differing  as  greatly  as 
lithium  and  mercury,  not  only  in  atomic  weight  but  in  other 
properties  as  well,  should  have  nearly  identical  atomic  heats.  It 
will  be  observed  that  the  atomic  heats  of  boron,  silicon,  carbon 
and  glucinum  are  too  low.  This  departure  from  the  law  of  Dulong 
and  Petit  is  more  apparent  than  real,  for  in  the  statement  of  the 
law  there  is  no  specification  as  to  the  temperature  at  which  the 
specific  heat  should  be  determined.  The  specific  heats  of  all 
solids  vary  with  the  temperature,  this  variation  being  greater  in 
the  case  of  some  elements  than  in  that  of  others.  It  has  been 
shown  that  the  specific  heats  of  the  above  four  elements  increase 
rapidly  with  rise  of  temperature  and  approach  limiting  values. 
As  these  values  are  approached  the  product  of  specific  heat  and 
atomic  weight  approximates  more  and  more  closely  to  the  mean 
value  of  the  constant,  6.4. 


12 


THEORETICAL  CHEMISTRY 


The  following  table  gives  the  values  obtained  by  Weber*  for 
carbon  and  silicon. 

VARIATION  OF  ATOMIC   HEAT  WITH  TEMPERATURE 

CARBON  (DIAMOND) 


Temperature, 
degrees. 

Sp.  Ht. 

At.  Ht. 

-50 

0.0635 

0.76 

+  10 

0.1128 

1.35 

85 

0.1765 

2.12 

206 

0.2733 

3.28 

607 

0.4408 

5.30 

806 

0.4489 

5.40 

985 

0.4589 

5.50 

CARBON  (GRAPHITE) 


Temperature, 
degrees. 

Sp.  Ht. 

At.  Ht. 

-50 

0.1138 

1.37 

+  10 

0.1604 

1.93 

61 

0.1990 

2.39 

202 

0.2966 

3.56 

642 

0.4454 

5.35 

822 

0.4539 

5.45 

978 

0.4670 

5.50 

SILICON 


Temperature, 
degrees. 

Sp.  Ht. 

At.  Ht. 

-40 

0.136 

3.81 

+57 

0.183 

5.13 

129 

0.196 

5.50 

232 

0.203 

5.63 

It  is  evident  that  this  empirical  relation  can  be  used  to  deter- 
mine the  approximate  atomic  weight  of  an  element  when  its 
specific  heat  is  known,  thus 

6.4 

atomic  weight  = ./   ,    -  •  (1) 

specific  heat 

The  law  of  Dulong  and  Petit  has  been  of  great  service  in  fixing 
and  checking  atomic  weights. 

About  twenty  years  after  the  law  of  Dulong  and  Petit  was 
*  Pogg.  Ann.,  154,  367  (1875). 


FUNDAMENTAL  PRINCIPLES 


13 


formulated,  Neumann*  showed  that  a  similar  relation  holds  for 
compounds  of  the  same  general  chemical  character.  Neumann's 
law  may  be  stated  thus:  Similarly  constituted  compounds  in  the 
solid  state  have  the  same  molecular  heat.  Subsequently  Koppf 
pointed  out  that  the  thermal  capacity  of  the  atoms  is  not  appreciably 
altered  when  they  enter  into  chemical  combination,  or  in  other  words, 
the  molecular  heat  of  solid  compounds  is  an  additive  property, 
being  made  up  of  the  atomic  heats  of  the  constituent  elements. 
For  example,  the  specific  heat  of  PbBr2  is  0.054  and  its  molec- 
ular weight  is  366.8;  therefore,  the  molecular  heat  is  0.054  X  366.8 
=  19.9.  Since  there  are  three  atoms  in  the  molecule,  19.9  -v-  3 
=  6.6  is  their  average  atomic  heat,  a  value  in  excellent  agree- 
ment with  the  constant  in  the  law  of  Dulong  and  Petit.  Neu- 
mann's law  may  be  used  to  estimate  the  atomic  heats  of  elements 
which  cannot  be  readily  investigated  in  the  solid  state.  The 
following  table  gives  a  list  of  atomic  heats  of  elements  in  the  solid 
state  derived  by  means  of  Neumann's  law. 


DERIVED  ATOMIC  HEATS 


Element. 

At.  lit. 

Element. 

At.  Ht. 

Hydrogen  

2.3 

Carbon  

1  8 

Oxygen 

4  0 

Silicon 

4  0 

Fluorine 

5  0 

Phosphorus 

5  4 

Nitrogen 

5.5 

Sulphur 

5  4 

Isomorphism.  From  a  study  of  the  corresponding  salts  of 
phosphoric  and  arsenic  acids,  MitscherlichJ  observed  that  they 
crystallize  with  the  same  number  of  molecules  of  water  and  are 
nearly  identical  in  crystalline  form,  it  being  possible  to  obtain 
mixed  crystals  from  solutions  containing  both  salts.  This  sug- 
gested to  Mitscherlich  a  line  of  investigation  which  resulted,  in 
1820,  in  the  establishment  of  the  law  of  isomorphism  which  bears 
his  name. 

This  law  may  be  stated  as  follows:  An  equal  number  of  atoms 
combined  in  the  same  manner  yield  the  same  crystal  form,  which  is 
independent  of  the  chemical  nature  of  the  atoms  and  dependent  upon 

*  Pogg.  Ann.,  23,  1  (1831). 

t  Lieb.  Ann.  (1864),  Suppl.,  3,  5. 

|  Ann.  Chim.  Phys.  (2),  14,  172  (1820). 


14  THEORETICAL  CHEMISTRY 

their  number  and  position.  Thus,  when  one  element  replaces 
another  in  a  compound  without  changing  its  crystalline  form, 
Mitscherlich  assumed  that  one  element  has  displaced  the  other, 
atom  for  atom.  For  example,  having  two  isomorphous  sub- 
stances, such  as  BaCl2.2  H2O  and  BaBr2.2  H2O,  we  assume  that 
the  bromine  in  the  second  compound  has  replaced  the  chlorine 
in  the  first  and,  if  the  atomic  weights  of  all  of  the  elements  in  the 
first  compound  are  known,  then  it  is  evident  that  the  atomic 
weight  of  the  bromine  in  the  second  compound  can  be  easily  cal- 
culated. This  method  was  largely  used  by  Berzelius  in  fixing 
atomic  weights  and  in  checking  the  values  obtained  by  the  vol- 
umetric method.  It  should  be  remembered  that  the  converse 
of  the  law  of  isomorphism  does  not  hold,  since  elements  may  re- 
place each  other,  atom  for  atom,  without  preserving  the  same 
form  of  crystallization.  Many  exceptions  to  the  law  have  been 
pointed  out.  For  example,  Mitscherlich  himself  showed  that 
Na2SO4  and  BaMn2Os  are  isomorphous  and  yet  the  two  mole- 
cules do  not  contain  the  same  number  of  atoms.  Furthermore, 
careful  measurements  of  the  interfacial  angles  of  crystals  have 
revealed  the  fact,  that  substances  which  have  been  regarded  as 
isomorphous  are  only  approximately  so ;  thus  the  interfacial  angles 
of  the  apparently  isomorphous  crystalline  salts  MgSO4.7  H2O, 
ZnSO4.7  H2O  and  NiS04.7  H2O  are  found  to  be  89°  26',  88°  53' 
and  88°  56',  respectively.  Ostwald  has  suggested  that  the  term 
homeomorphous  be  applied  to  designate  substances  which  have 
nearly  identical  form.  At  best  the  principle  of  isomorphism 
is  only  an  approximation  and  should  be  employed  with  caution. 
Valence.  During  the  latter  half  of  the  nineteenth  century 
the  usefulness  of  the  atomic  theory  was  greatly  enhanced  by.  the 
introduction  of  certain  assumptions  concerning  the  combining 
power  of  the  atoms.  These  assumptions,  constituting  the  so- 
called  doctrine  of  valence,  were  forced  upon  chemists  in  order 
that  a  satisfactory  explanation  might  be  offered  of  the  phenom- 
enon of  isomerism.  A  consideration  of  the  following  formulas,  - 
HC1,  H20,  NH3,  CH4,  —  shows  that  the  power  to  combine  with 
hydrogen  increases  regularly  from  chlorine,  which  combines  with 
hydrogen,  atom  for  atom,  to  carbon,  one  atom  of  which  is  capable 
of  combining  with  four  atoms  of  hydrogen.  Either  hydrogen  or 
chlorine,  each  of  which  is  capable  of  combining  with  but  one 
atom  of  the  other,  may  be  taken  as  an  example  of  the  simplest 


FUNDAMENTAL  PRINCIPLES  15 

kind  of  atom.  Any  element  like  hydrogen  or  chlorine  is  called 
a  univalent  element,  whereas  elements  similar  to  oxygen,  nitro- 
gen and  carbon,  which  are  capable  of  combining  with  two,  three 
or  four  atoms  of  hydrogen,  are  called  bivalent,  trivalent  and  quadri- 
valent elements  respectively.  Most  elements  belong  to  one  or 
the  other  of  these  four  classes,  although  quinquivalent,  sexivalent 
and  septivalent  elements  are  known.  The  familiar  bonds  or  link- 
ages of  structural  formulas  are  graphic  representations  of  the 
valence  of  the  atoms  constituting  the  molecule.  This  useful  con- 
ception of  valence  has  made  possible  the  prediction  of  the  prop- 
erties of  many  compounds  before  they  have  been  discovered  in 
nature  or  in  the  laboratory. 

Atomic  Weights.  Among  the  first  to  recognize  the  importance 
of  Dalton's  atomic  theory  was  the  Swedish  chemist,  Berzelius. 
He  foresaw  the  importance  for  chemists  of  a  table  of  exact  atomic 
weights,  and  in  1810  he  undertook  the  task  of  determining  the 
combining  weights  of  most  of  the  known  elements.  For  nearly 
six  years  he  was  engaged  in  determining  the  exact  composition  of 
a  large  number  of  compounds  and  calculating  the  combining 
weights  of  K  their  constituent  elements,  thus  compiling  the  first 
table  of  atomic  weights. 

Numerous  investigators  since  Berzelius  have  been  engaged  in 
this  important  work,  among  whom  should  be  mentioned  Stas, 
Marignac,  Morley  and  Richards.  On  two  occasions  special  stim- 
ulus was  given  to  such  investigations.  The  first  occasion  was  in 
1815,  when  Prout  suggested  that  the  atomic  weights  of  the  ele- 
ments are  exact  multiples  of  the  atomic  weight  of  hydrogen.  The 
values  obtained  by  Berzelius  were  incompatible  with  the  hypoth- 
esis of  Prout,  although  the  atomic  weights  of  several  of  the  ele- 
ments differed  but  little  from  integral  values.  To  test  the  accu- 
racy of  'this  hypothesis,  Stas  undertook  the  determination  of  the 
atomic  weights  of  several  elements  with  a  degree  of  accuracy 
such  that  his  maximum  experimental  error  was  less  than  the  differ- 
ence between  the  atomic  weight  found  and  the  nearest  whole 
number.  This  important  series  of  investigations  disproved  Prout's 
hypothesis  as  originally  stated.  The  second  occasion  when  the 
investigation  of  atomic  weights  received  a  special  impulse,  was  in 
1869,  when  Mendeleeff  brought  forward  the  periodic  classification 
of  the  elements.  When  the  elements  were  arranged  in  the  order 
of  their  atomic  weights,  several  of  them  were  found  to  fall  in  groups 


16  THEORETICAL  CHEMISTRY 

with  which  their  chemical  and  physical  properties  did  not  corre- 
spond, and  Mendeleeff  asserted  that  in  these  cases  the  commonly 
accepted  atomic  weights  were  erroneous.  This  led  to  the  careful 
redetermination  of  the  atomic  weights  which  Mendeleeff  had 
asserted  to  be  faulty,  and  in  most  cases  his  predictions  were  con- 
firmed. 

In  connection  with  the  precise  determination  of  atomic  weights, 
the  work  of  Cannizzaro  in  the  latter  part  of  the  nineteenth  century 
should  be  mentioned.  He  emphasized  the  importance  of  Avoga- 
dro's  law  as  the  basis  of  atomic  weight  determinations,  and  drew 
a  sharp  (distinction  between  atomic  and  molecular  weights,  thus 
bringing  order  out  of  confusion  and  rendering  possible  the  present 
system  of  atomic  weights. 

Determination  of  Atomic  Weights  There  are  two  funda- 
mentally different  methods  for  the  precise  determination  of  atomic 
weights;  one  based  upon  stoichiometry  and  chemical  change, 
and  the  other  upon  the  principle  of  Avogadro  and  the  deviations 
from  the  laws  of  Boyle  and  Gay  Lussac.  The  former  is  com- 
monly referred  to  as  the  chemical  method,  in  contradistinction 
to  the  latter  which  is  known  as  the  physical  method. 

During  the  last  twenty-five  years,  the  degree  of  precision  at- 
tainable in  the  determination  of  atomic  weights  by  the  chemical 
method  has  been  greatly  increased,  notably  by  the  introduction 
of  the  numerous  refinements  in  manipulation  developed  by  T. 
W.  Richards.  The  atomic  weight  determinations  carried  out  by 
Richards  and  his  associates  are  universally  regarded  as  master- 
pieces of  experimental  work,  and  it  is  to  these  investigations  that 
we  are  largely  indebted  for  the  commonly  accepted  values  of  the 
atomic  weight  of  at  least  twenty  of  the  elements.  The  following 
example,  taken  from  a  paper  by  Richards,  Kothner  and  Tiede* 
on  the  determination  of  the  atomic  weights  of  nitrogen,  chlorine 
and  silver,  affords  an  illustration  of  the  method  of  determining 
and  calculating  atomic  weights  by  the  chemical  method.  The 
weight  of  silver  chloride  obtainable  from  an  accurately  known 
weight  of  ammonium  chloride,  in  the  reaction  represented  by  the 
equation 

AgNO3  +  NH4C1  =  AgCl  +  NH4N03, 

was  determined,  every  precaution  being  taken  to  insure  the  purity 

of  the  reacting  materials.     The  average  value  of  the  ratio  of 

*  Jour.  Am.  Chem.  Soc.  31,  17  (1909). 


FUNDAMENTAL  PRINCIPLES  17 

the  weights  of  silver  chloride  to  ammonium  chloride,  as  calculated 
from  nine  different  determinations,  was  found  to  be  0.373217. 
From  this  result  the  atomic  weights  of  nitrogen,  chlorine  and 
silver  can  be  calculated  as  follows: 

Let  AgCl/Ag  =  a,  (1) 

NH4Cl/AgCl  =  b,  (2) 

and  AgNOa/Ag  =  c.  (3) 

The  three  atomic  weights  supposed  to  be  unknown  may  be  desig- 
nated as  follows:  Ag  =  x,  Cl  =  y  and  N  =  z.  The  value  of 
the  atomic  weight  of  hydrogen  is  taken  as  1.0076  on  the  basis  of 
0  =  16.  Substituting  these  values  in  equations  (1),  (2),  and 
(3),  we  obtain  the  following: 

x  +  y  =  ax,  (4) 

z  +  y  +  4.0304  =  b~  (x  +  y),  (5) 

and  x  +  z  +  48.000  =  ex.  (6) 

Substituting  in  (5)  the  value  of  z  as  found  from  equation  (6), 
we  have: 

(1  -  c)  x-bx  +  (l-b)  y  =  43.9696, 


but  according  to  (4)  y  =  (a  —  1)  x\  hence 

43.9696 


1  -  c  -  b  +  (1  -  6)  (a  -  1) 


(7) 


The  values  a,  b,  and  c  are  all  known,  a  (the  quantity  of  silver 
chloride  obtained  from  one  gram  of  silver)  was  found  by  Richards 
and  Weljs*  to  be  1.32867;  c  (the  quantity  of  silver  nitrate  ob- 
tained from  the  same  quantity  of  silver)  was  found  by  Richards 
and  Forbes  f  to  be  1.57479.  The  value  of  6  is  given  as  the  aver- 
age of  the  ratio  of  the  weights  of  silver  chloride  to  ammonium 
chloride,  or  0.373217.  On  substituting  these  values  in  (7),  we 
obtain  x,  the  atomic  weight  of  silver,  =  107.881.  Substituting 
this  value  in  (4),  we  obtain  y,  the  atomic  weight  of  chlorine,  = 
35.4574,  and  substituting  these  values  in  (5)  or  (6),  we  obtain 
z,  the  atomic  weight  of  nitrogen,  =  14.0085. 

The  physical  method  for  the  precise  determination  of  atomic 
weights  is  based  upon  Avogadro's  principle.     While  this  prin- 

*  Publ.  Carnegie  Inst.,  No.  28. 
f  Publ.  Carnegie  Inst.,  No.  69. 


18  THEROETICAL  CHEMISTRY 

ciple  is  known  to  be  strictly  true  for  ideal  gases  only,  it  has  been 
shown  by  D.  Berthelot  and  others  that,  if  due  allowance  is  made 
for  the  compressibility  of  a  gas,  the  experimentally  determined 
value  of  its  density  may  be  corrected  so  as  to  render  possible 
the  calculation  of  the  exact  value  of  the  molecular  weight  of  the 
gas.  Making  use  of  this  so-called  method  of  "  limiting  densities," 
Gray*  has  determined  the  atomic  weight  of  nitrogen  with  great 
precision.  Instead  of  using  nitrogen  itself,  however,  he  found  it 
more  convenient  to  employ  nitric  oxide.  This  gas  was  prepared 
in  a  high  state  of  purity  and  its  density  determined  with  the  ut- 
most care.  The  molecular  weight  of  NO,  as  calculated  from  the 
corrected  value  of  the  density,  was  found  to  be  30.004;  if  we  de- 
duct from  this  the  value  of  the  atomic  weight  of  oxygen,  we  have 
30.004  -  16.000  =  14.004  as  the  value  of  the  atomic  weight  of 
nitrogen. 

The  relative  accuracy  of  the  two  methods  for  the  determination 
of  atomic  weights  has  been  carefully  considered  by  Guyef  who 
concludes  that  the  results  obtained  by  the  physical  method  compare 
favorably  with  those  obtained  by  the  purely  chemical  method. 

International  Atomic  Weights.  Dalton  selected  hydrogen, 
the  lightest  known  element,  as  the  unit  of  his  system  of  combin- 
ing weights  of  the  elements,  but  Berzelius  pointed  out  that  this 
was  an  unwise  choice  since  but  relatively  few  of  the  elements  form 
stable  compounds  with  hydrogen.  He  proposed,  therefore,  that 
oxygen  should  be  taken  as  the  standard,  assigning  to  it  the  arbi- 
trary value  100.  This  proposal  of  Berzelius,  to  substitute  oxygen 
for  hydrogen  as  the  unit  of  atomic  weights,  did  not  receive  serious 
consideration  until  quite  recently,  when  the  International  Com- 
mittee on  Atomic  Weights  took  up  the  matter  and,  after  careful 
deliberation,  decided  in  favor  of  the  oxygen  standard.  The 
atomic  weight  of  oxygen  is  taken  as  16,  and  the  unit  to  which  all 
atomic  weights  are  referred  is  one-sixteenth  of  this  weight.  The 
atomic  weight  of  hydrogen  on  this  basis  is  1.008.  Aside  from  the 
fact  that  most  of  the  elements  form  compounds  with  oxygen  which 
are  suitable  for  analysis,  the  atomic  weights  of  more  of  the  ele- 
ments approximate  to  integral  values  when  oxygen  instead  of  hy- 
drogen is  used  as  the  standard. 

The  table  on  page  19  gives  the  values  of  the  atomic  weights 

*  Jour.  Chem.  Soc.  87,  1601  (1905). 
t  Jour  Chim.  Phys.  3,  352  (1905). 


FUNDAMENTAL  PRINCIPLES 


19 


as  published  by  the  International  Committee  on  Atomic  Weights 
for  1922. 

1922 
INTERNATIONAL  ATOMIC  WEIGHTS 


Aluminium.  .  . 

Al 

O  =  16 

27.1 

Molybdenum.  .  . 

Mo 

Qfi  0 

Antimony.  .  .  . 

Sb 

120.2 

Neodymium  

Nd 

144  3 

Argon  

A 

39.90 

Neon  

Ne 

20  2 

Arsenic  

As 

74.96 

Nickel.  . 

Ni 

CvQ    fiQ 

Barium 

Ba 

137  37 

Niton  (radium  emanation) 

ooo  4. 

Bismuth  

Bi 

208.0 

Nitrogen  

N 

14  008 

Boron  

B 

10.9 

Osmium  

Os 

190  9 

Bromine  

Br 

79.92 

Oxygen  

o 

16  00 

Cadmium  

Cd 

112.40 

Palladium 

Pd 

106  7 

Caesium  

Cs 

132.81 

Phosphorus     .  . 

p 

31  04 

Calcium  

Ca 

40.07 

Platinum  

Pt 

195  2 

Carbon 

c 

12  005 

Potassium 

K 

39  10 

Cerium  . 

Ce 

140  25 

Praseodymium 

Pr 

140  9 

Chlorine  

Cl 

35.46 

Radium 

Ra 

226  0 

Chromium  

Cr 

52.0 

Rhodium. 

Rh 

102  9 

Cobalt  

Co 

58.97 

Rubidium  

Rb 

85  45 

Columbium.  .  .  . 

Cb 

93.1 

Ruthenium  

Ru 

101  7 

Copper.  . 

Cu 

63.57 

Samarium  

Sa 

150  4 

Dy  spr  os  i  um  . 

Dy 

162  5 

Scandium 

Sc 

44  1 

Erbium 

F,r 

167  7 

Selenium 

Se 

79  2 

Europium  

Eu 

152  0 

Silicon 

Si 

28  3 

Fluorine  

F 

19  0 

Silver.  .  . 

Ag 

107  88 

Gadolinium.  .  .  . 

Gd 

157.3 

Sodium  

Na 

23  00 

Gallium  

Ga 

70.1 

Strontium  

Sr 

87  63 

Germanium.  .  .  . 

Ge 

72.5 

Sulphur  

S 

32.06 

Glucinum 

Gl 

9  1 

Tantalum 

Ta 

181  5 

Gold     

Au 

197  2 

Tellurium 

Te 

127  5 

Helium  

He 

4  00 

Terbium 

Tb 

159  2 

Holmium  

Ho 

163  5 

Thallium. 

Tl 

204  0 

Hydrogen 

H 

1  008 

Thorium 

Th 

232  15 

Indium 

In 

114  8 

Thulium 

Tm 

168  5 

Iodine 

I 

126  92 

Tin 

Sn 

118  7 

Iridium 

Ir 

193  1 

Titanium 

Ti 

48  1 

Iron 

Fe 

55  84 

Tungsten 

W 

184  0 

Krypton      .... 

Kr 

82  92 

Uranium                         

u 

238.2 

Lanthanum.  .  . 

L/a 

139  0 

Vanadium.                 

V 

51.0 

Lead  

Pb 

207  20 

Xenon    

Xe 

130.2 

Lithium  
Lutecium 

L,i 
Lu 

6.94 
175  0 

Ytterbium  (Neoytterbium) 
Yttrium 

Yb 

Yt 

173.5 
89  33 

Magnesium 

Vler 

24  32 

Zinc                         

Zn 

65.37 

Manganese  .... 

AT 

Yin 

54  93 

Zirconium         

Zr 

90.6 

Mercury  

Hg 

200  6 

REFERENCES 

Stoichiometry.     Sidney  Young. 

On  the  Densities  of  Hydrogen  and  Oxygen,  and  on  the  Ratio  of  their  Atomic 

Weights.     E.  W.  Morley.     (Smithsonian  Contributions  to  Knowledge, 

No.  980.) 
Les  Travaux  de  1'Universite'  de  Harvard  sur  les  Poids  Atomiques.     T.  W. 

Richards.    Journal  de  Chimie  Physique,  6,  92  (1908). 


20  THEORETICAL  CHEMISTRY 

PROBLEMS 

1.  Stas  found  on  heating  498.6355  grams  of  potassium  chlorate  that 
the  residue  of  potassium  chloride  weighed  303.3870  grams.     Assuming 
that  the  atomic  weights  of  oxygen  and  chlorine  are  16  and  35.46  respec- 
tively, calculate  the  value  of  the  atomic  weight  of  potassium. 

2.  On  burning  16.1920  grams  of  pure  carbon,  59.3765  grams  of  carbon 
dioxide  is  obtained.     If  the  atomic  weight  of  oxygen  is  taken  as  16,  what 
is  the  value  of  the  atomic  weight  of  carbon? 

3.  Calculate  the  value  of  the  atomic  weight  of  barium  from  the  follow- 
ing data:   100  grams  of  barium  chloride  yield  112.09  grams  of  barium  sul- 
phate; 0  =  16,  S  =  32.06,  Cl  =  35.46. 

4.  Erdmann  and  Marchand  found  on  heating  13.6031  grams  of  cal- 
cium carbonate  that  the  residue  of  calcium  oxide  weighed  7.6175  grams. 
What  is  the  value  of  the  atomic  weight  of  calcium?      (0  =  16,  C  =  12). 

5.  The  atomic  weight  of  silver  is  107.88  and  its  specific  heat  is  0.057. 
The  specific  heat  of  a  metal  M  is  found  to  be  0.0306  and  the  analysis  of 
its  chloride  shows  that  70  parts  of  the  metal  unite  with  35.46  parts  of 
chlorine.     Find  its  atomic  weight. 

6.  The  specific  heat  of  cadmium  is  0.0567  and  its  equivalent  is  56. 
Calculate  its  atomic  weight. 

7.  Baxter  and  Grover  in  their  determination  of  the  atomic  weight  of 
lead  found  the  ratio  PbCl2/2Ag  =  1.28905.     If  the  atomic  weights  of 
silver  and  chlorine  are  107.880  and  35.457  respectively,  calculate  the 
value  of  the  atomic  weight  of  lead. 

8.  The  weight  of  1  liter  of  carbon  dioxide  as  computed  from  the  value 
of  the  limiting  density  is  1.9635  grams.     Calculate  the  value  of  the  atomic 
weight  of  carbon  on  the  basis  of  0  =  16. 

9.  The  weight  of  1  liter  of  sulphur  dioxide  as  computed  from  the  value 
of  its  limiting  density  is  2.8586  grams.    Calculate  the  value  of  the  atomic 
weight  of  sulphur. 

10.  The  specific  gravity  of  ammonia  referred  to  oxygen  is  0.53940. 
Calculate  the  approximate  value  of  the  atomic  weight  of  nitrogen,  hav- 
ing given  the  atomic  weight  of  hydrogen,  1.008. 


CHAPTER  II 
GASES 

The  Gas  Laws.  Matter  in  'the  gaseous  state  possesses  the 
property  of  filling  completely  and  to  a  uniform  density  any  avail- 
able space.  Among  the  most  pronounced  characteristics  of  gases 
are  lack  of  definite  shape  or  volume,  low  density  and  small  vis- 
cosity. The  laws  expressing  the  behavior  of  gases  under  differ- 
ent conditions  are  relatively  simple  and  to  a  large  extent  are 
independent  of  the  nature  of  the  gas.  The  temperature  and 
pressure  coefficients  of  all  gases  are  very  nearly  the  same. 

In  1662,  Robert  Boyle  discovered  the  familiar  law  that  at 
constant  temperature,  the  volume  of  a  gas  is  inversely  proportional 
to  the  pressure.  This  may  be  expressed  mathematically  as  fol- 
lows:— 

v  oc  -  (temperature  constant) 

where  v  is  the  volume  and  p  the  pressure. 

In  1801,  Gay-Lussac  discovered  the  law  of  the  variation  of  the 
volume  of  a  gas  with  temperature. 

This  law  may  be  formulated  thus:  —  At  constant  pressure,  the 
volume  of  a  gas  is  directly  proportional  to  its  absolute  temperature, 
or 

vcc  T  (pressure  constant). 

There  are  three  conditions  which  may  be  varied,  viz.,  volume, 
temperature  and  pressure.  The  preceding  laws  have  dealt  with 
the  relation  between  two  pairs  of  the  variables  when  the  third 
is  held  constant.  There  remains  to  consider  the  relation  between 
the  third  pair  of  variables,  pressure  and  temperature,  the  volume 
being  kept  constant.  Evidently  a  necessary  corollary  of  the  first 
two  laws  is,  that  at  constant  volume,  the  pressure  of  a  gas  is  directly 
proportional  to  its  absolute  temperature,  or 

p  oc  T  (volume  constant). 
21 


22  THEORETICAL  CHEMISTRY 

These  three  laws  may  be  combined  into  a  single  mathematical 
expression  as  follows  :  — 

v  oc  -  (T  const.)  law  of  Boyle, 
v  oc  T  (p  const.)  law  of  Gay-Lussac; 
combining  these  two  variations  we  have, 

T 

V  OC   —) 

p 

or  introducing  a  proportionality  factor  fc, 

7    T 

v  =  K  —  J 

P 
or 

vp  =  kT.  (1) 

If  the  temperature  of  the  gas  be  0°  (273°  absolute),  and  the  corre- 
sponding volume  and  pressure,  v0  and  po  respectively,  then  (1) 

becomes 

v0pQ  =  273  k, 

and 


f  2) 
~273' 

eliminating  the  constant  k  between  (1)  and  (2),  we  have 


For  any  one  gas  the  term  7       is  a  constant.     If  ^o  is  the  volume 

^7o 

of  1  gram  of  gas  at  0°  and  76  cm.,  we  write 

vp  =  rT,  (3) 

where  v  is  the  volume  of  1  gram  of  gas  at  the  temperature  T  and 
the  pressure  p,  and  r  is  a  constant  called  the  specific  gas  constant. 
On  the  other  hand  when  VQ  denotes  the  volume  of  one  mol  of  gas 
at  0°  and  76  cm.  (22.4  liters),  the  equation  becomes 

vp  =  RT,  (4) 

where  R  is  termed  the  molar  gas  constant,  which  has  the  same 
value  for  all  gases.  If  M  is  the  molecular  weight  of  the  gas,  Mr 
=  R.  Equation  (4)  is  the  fundamental  gas  equation 


GASES  23 

Evaluation  of  the  Molar  Gas  Constant.  Since  the  product 
of  p  and  v  represents  work,  and  T  is  a  pure  number,  R  must  be 
expressed  in  energy  units.  There  are  four  different  units  in  which 
the  molar  gas  constant  is  commonly  expressed,  viz.,  (1)  gram- 
centimeters,  (2)  ergs,  (3)  calories,  and  (4)  liter-atmospheres. 

1.  R  in  gram-centimeters.     The  volume,  v,  of  1  mol  of  gas  at 
0°  and  76  cm.  is  22.4  liters  or  22,400  cc.     The  pressure,  p,  is  76  cm. 
multiplied  by  13.59,  (the  density  of  mercury),  or  1033.3  grams  per 
square  centimeter.     Substituting  we  obtain 

PQVQ      1033.3  X  22,400 
R  =  ^-  =  ~         273         -  =  84,760  gr.  cm. 

2.  R  in  ergs.     To  convert  gram-centimeters  into  ergs  we  must 
multiply  by  the  acceleration  due  to  gravity,  g  =  980.6  cm.  per 
sec.  per  sec.,  or 

R  =  84,760  X  980.6  =  83,150,000  ergs. 

3.  R  in  calories.     To  express  work  in  terms  of  heat,  we  must 
divide  by  the  mechanical  equivalent  of  heat,  or,  since  1  calorie  is 
equivalent  to  42,640  gr.  cm.  or  41,830,000  ergs,  we  have 


R  =  -  1-99  cal.  (approximately  2  cal.) 


4.  R  in  liter-atmospheres.  A  liter-atmosphere  may  be  denned 
as  the  work  done  by  1  atmosphere  on  a  square  decimeter  through 
a  decimeter.  If  p0  is  the  pressure  in  atmospheres,  and  v0  is  the 
volume  in  liters,  we  have 


R  =         =          =  0.0821  liter-atmosphere. 
1 


Avogadro's  Constant.  Several  recent  experimental  investi- 
gations have  furnished  the  necessary  data  for  the  calculation 
of  the  actual  number  of  molecules  contained  in  one  gram-molecule 
of  gas.  This  number  has  been  called  the  Avogadro  constant 
or  Avogadro  number.  Notwithstanding  the  fact  that  a  wide 
variety  of  experimental  methods  now  exist  upon  which  inde- 
pendent calculations  of  this  constant  can  be  based,  it  is  interesting 
to  note  that  the  agreement  of  the  results  obtained  by  these  differ- 
ent methods  is  exceedingly  close.  All  of  the  methods  which  have 
been  employed  in  the  evaluation  of  Avogadro's  constant  have 


24 


THEORETICAL  CHEMISTRY 


been  subjected  to  the  most  rigorous  tests  to  insure  their  accuracy 
together  with  that  of  the  calculations  based  thereon. 

The  following  table  gives  a  partial  list  of  the  values  of  the  Avo- 
gadro  constant  as  determined  by  various  observers  using  different 

VALUE  OF  AVOGADRO'S  CONSTANT 


Observer. 

Method. 

No.    Molecules    in 
One  Gram-molecule 
=  N  X  lO2" 

Perrin 

Brownian  movement  

6.8 

Millikan 

Determination   of  elemental 

electric  charge 

6  06  ±  0  001 

Ruth6rford 

Charge  on    -particle    , 

6  2 

Boltwood 

a-particles  from  radium 

6.3 

Curie 

a-particles  from  polonium.  .  .  . 

6.5 

King 

Rate  of  fall  of  oil  drops  

6.03 

Planck  .  .  . 

Radiation  

6.06 

experimental  methods.  Of  these  different  values,  undoubtedly 
that  due  to  Millikan  in  which  the  estimated  error  does  not  exceed 
0.1  per  cent  is  the  most  trustworthy. 

It  has  been  found  that,  under  the  most  favorable  conditions, 
the  degree  of  rarefaction  obtainable  with  a  modern  high-vacuum 
pump  is  such  that  the  number  of  molecules  per  cubic  centimeter  is 
reduced  to  approximately  7  X  109.  When  this  figure  is  compared 
with  2.705  X  1019,  the  number  of  molecules  present  in  one  cubic 
centimeter  of  gas,  the  futility  of  attempting  to  produce  a  per- 
fect vacuum  is  quite  obvious. 

Deviations  from  the  Gas  Laws.  Careful  experiments  by  Ama- 
gat*  and  others  on  the  behavior  of  gases  over  extended  ranges 
of  temperature  arid  pressure  have  shown  that  the  fundamental 
gas  equation,  pv  =  RT,  is  not  strictly  applicable  to  any  one  gas, 
the  deviations  depending  upon  the  nature  of  the  gas  and  the 
conditions  under  which  it  is  observed.  It  has  been  shown  that 
the  gas  laws  are  more  nearly  obeyed  the  lower  the  pressure,  the 
higher  the  temperature  and  the  further  the  gas  is  removed  from 
the  critical  state.  A  gas  which  would  conform  to  the  require- 
ments of  the  fundamental  gas  equation  is  called  an  ideal  or  per- 
fect gas.  Almost  all  gases  are  far  from  ideal  in  their  behavior. 

*  Ann.  Chim.  phys.  (5)  19,  345  (1880). 


GASES 


25 


At  constant  temperature  the  product,  pv,  in  the  gas  equation  is 
constant,  so  that  if  we  plot  pressures  as  abscissae  and  the  corre- 
sponding values  of  pv  as  ordinates,  for  an  ideal  gas  we  should 


Hydrogen 


20     40      60      80     100    120    140    160    180    200    220    240    260    280    300 
p  in  meters  of  Mercury 

Fig.  2 

obtain  a  straight  line  parallel  to  the  axis  of  abscissae.  The  results 
obtained  with  six  typical  gases  are  also  shown  in  the  accompanying 
diagrams  Figs.  2  to  5.  It  will  be  apparent  that  all  of  these  gases 


Nitrogen 


s, 


40  80  120          160  200          240          280          320 

p  in  meters  of  Mercury 

Fig.  3 

depart  widely  from  ideal  behavior.  In  the  case  of  hydrogen  and 
neon  pv  increases  continuously  with  the  pressure,  while  with  nitro- 
gen and  carbon  dioxide  it  first  decreases,  attains  to  a  minimum 
value  and  then  increases  with  increasing  pressure.  Oxygen  gives 


26 


THEORETICAL  CHEMISTRY 


a  straight  line  sloping  upwards  to  the  pv  axis,  while  helium  gives 
a  perfectly  horizontal  line.     Although  helium  thus  conforms  to 


tOO  200 

P  in  meters  of  Mercury 

Fig.  4 


I, 


Helium  (0°) 


250  350  450  550  650 

p  in  m.m.  of  Mercury 

Fig.  5 


750 


850 


Boyle's  law  at  0°,  it  fails  to  behave  ideally  at  higher  temperatures. 
With  the  exception  of  hydrogen,  all  gases  show  a  minimum  in 
the  curve,  thus  indicating  that  at  first  the  compressibility 
is  greater  than  corresponds  with  the  law  of  Boyle,  but  diminishes 


GASES  27 

continuously  until,  for  a  short  range  of  pressure,  the  law  is  followed 
strictly;  beyond  this  point  the  compressibility  is  less  than  Boyle's 
law  requires. 

Hydrogen  is  exceptional  in  that  it  is  always  less  compressible 
than  the  law  demands.  This  is  true  for  all  ordinary  tempera- 
tures, but  it  is  highly  probable  that  at  extremely  low  temperatures 
the  curve  would  show  a  minimum.  The  two  curves  for  carbon 
dioxide  at  31°. 5  and  100°  illustrate  the  fact  that  the  deviations 
from  the  gas  laws  become  less  as  the  temperature  increases.  The 
deviations  of  gases  from  the  laws  of  Boyle  and  Gay-Lussac,  as  well 
as  their  behavior  in  general,  may  be  satisfactorily  accounted  for 
on  the  basis  of  the  kinetic  theory. 

Kinetic  Theory  of  Gases.  The  first  attempt  to  explain  the 
properties  of  gases  on  a  purely  mechanical  basis  was  made  by 
Bernoulli  in  1738.  Subsequently,  through  the  labors  of  Kroenig, 
Clausius,  Maxwell,  Boltzmann  and  others,  his  ideas  were  devel- 
oped into  what  is  known  today  as  the  kinetic  theory  of  gases. 
According  to  this  theory,  gases  are  considered  to  be  made  up  of 
minute,  perfectly  elastic  particles  which  are  ceaselessly  moving 
about  with  high  velocities,  colliding  with  each  other  and  with  the 
walls  of  the  containing  vessel.  These  particles  are  identical  with 
the  molecules  defined  by  Avogadro.  The  volume  actually  occu- 
pied by  the  gas  molecules  is  supposed  to  be  much  smaller  than  the 
volume  filled  by  them  under  ordinary  conditions,  thus  allowing 
the  molecules  to  move  about  free  from  one  another's  influence 
except  when  they  collide.  The  distance  through  which  a  mole- 
cule moves  before  colliding  with  another  molecule  is  known  as  its 
mean  free  path.  In  terms  of  this  theory,  the  pressure  exerted  by  a 
gas  is  due  to  the  combined  effect  of  the  impacts  of  the  moving 
molecules  upon  the  walls  of  the  containing  vessel,  the  magnitude 
of  the  pressure  being  dependent  upon  the  kinetic  energy  of  the 
molecules  and  their  number. 

Derivation  of  the  Kinetic  Equation.  Starting  with  the  assump- 
tions already  made,  it  is  possible  to  derive  a  formula  by  means  of 
which  the  gas  laws  may  be  deduced.  Imagine  n  molecules,  each 
having  a  mass,  m,  confined  within  the  cubical  vessel  shown  in 
Fig.  6,  the  edge  of  which  has  a  length,  I.  While  the  different 
molecules  are  doubtless  moving  with  different  velocities,  there 
must  be  an  average  velocity  for  all  of  them.  Let  u  denote  this 
mean  velocity  of  translation.  The  molecules  will  impinge  upon 


28 


THEORETICAL  CHEMISTRY 


the  walls  in  all  directions  but  the  velocity  of  each  may  be  resolved 
according  to  the  well-known  dynamical  principle  into  three  com- 
ponents, x,  y  and  z,  parallel  to  the  three  rectangular  axes,  X,  Y 
and  Z.  The  analytical  expression  for  the  velocity  of  a  single 
molecule,  M,  is 

u2  =  x2  +  y2  +  z2. 

In  words,  this  means  that  the  effect  of  the  collision  of  the  mole- 
cule upon  the  wall  of  the  containing  vessel,  is  equivalent  to  the 

combined  effect  of  suc- 
cessive collisions  of  the 
molecule  perpendicular 
to  the  three  walls  of  the 
cubical  vessel  with  the 
velocities  x,  y  and  z 
respectively.  Fixing 
our  attention  upon  the 
horizontal  component, 
the  molecule  will  collide 
with  the  wall  with  a 
velocity  x,  and  owing  to 
its  perfect  elasticity  it 
will  rebound  with  a 


Fig.  6 


velocity  —  x,  having  suffered  no  loss  in  kinetic  energy.  The  mom- 
entum before  collision  was  mx  and  after  collision  it  will  be  —  mx, 
the  total  change  in  momentum  being  2  mx.  The  distance  between 
the  two  walls  being  /,  the  number  of  collisions  on  a  wall  in  unit 
time  will  be,  x/l,  and  the  total  effect  of  a  single  molecule  in  one 
direction  in  unit  time  will  be  2  mx  •  x/l  =  2  mx2/l.  The  same 
reasoning  is  applicable  to  the  other  components,  so  that  the  com- 
bined action  of  a  single  molecule  on  the  six  sides  of  the  vessel 
will  be 


O     777  7?  7/^ 

There  being  n  molecules,  the  total  effect  will  be  -  — The 

entire  inner  surface  of  the  cubical  vessel  being  6  I2,  the  pressure 
p,  on  unit  area,  will  be 

_  2  mnu2  _  I   mnu2 

c  £}    7S  ~f)  73 


GASES  29 

But  since  I3  is  the  volume  of  the  cube,  which  we  will  denote  by  v, 
we  have 

1    mnu2 


or 

pv  =  ^mnu2.  (5) 

This  is  the  fundamental  equation  of  the  kinetic  theory  of  gases. 
While  the  equation  has  been  derived  for  a  cubical  vessel,  it  is 
equally  applicable  to  a  vessel  of  any  shape  whatever,  since  the 
total  volume  may  be  considered  to  be  made  up  of  a  large  number 
of  infinitesimally  small  cubes,  for  each  of  which  the  equation  holds. 

Deductions  from  the  Kinetic  Equation.  Law  of  Boyle.  In 
the  fundamental  kinetic  equation,  pv  =  \  mnu2,  the  right-hand 
side  is  composed  of  factors  which  are  constant  at  ^6nstant  temper- 
ature, and  therefore  the  product,  pv,  must  be  constant  also  under 
similar  conditions.  This  is  clearly  Boyle's  law. 

Law  of  Gay-Lussac.  The  kinetic  equation  may  be  written  in 
the  form 

2   1 
pv  =-5*0  mnu2.  (6) 

o    & 

The  kinetic  energy  of  a  single  molecule  being  represented  by 
1/2  mu2}  the  total  kinetic  energy  of  the  molecules  of  the  gas  will 
be  1/2  mnu2.  Therefore  the  product  of  the  pressure  and  volume  of 
the  gas  is  equal  to  two-thirds  of  the  total  kinetic  energy  of  its  molecules. 
Since  the  mean  kinetic  energies  of  all  gases  are  identical  at  the 
same  temperature,  it  follows  that  any  change  in  the  kinetic  energy 
resulting  from  a  change  in  temperature  must  be  directly  propor- 
tional to  the  latter,  otherwise  the  mean  kinetic  energies  of  the  gas 
molecules  which  were  equal  at  one  temperature  could  not  be  so  at 
any  other.  But  equation  (6)  teaches  that  the  product  of  the  pres- 
sure and  volume  of  a  gas  is  directly  proportional  to  the  total 
kinetic  energy  of  its  molecules,  and  a  change  in  temperature  must 
therefore  alter  this  product  to  the  same  extent  for  each  gas.  If  the 
pressure  be  maintained  constant,  the  volume  alone  will  undergo 
change,  or  vice  versa,  if  the  volume  remains  constant  the  pressure 
only  will  be  altered.  In  either  case,  whichever  factor  of  the  prod- 
uct, pv,  undergoes  change,  the  kinetic  theory  shows  that  the  rate 


30  THEORETICAL  CHEMISTRY 

of  this  change  with  temperature  is  the  same  for  all  gases.  This, 
it  will  be  observed  is  a  statement  of  the  law  of  Gay-Lussac. 

Hypothesis  of  Avogadro.     If  equal  volumes  of  two  different  gases 
are  measured  under  the  same  pressure,  we  will  have 

pv  =  l/3ttirai^i2  =  l/3n2ra2tt22,  (7) 

where  n\  and  n2,  mi  and  w2,  and  u\  and  u%  denote  the  number,  mass 
and  velocity  6f  the  molecules  in  the  two  gases.  If  the  gases  are 
measured  at  the  same  temperature,  the  molecules  of  each  possess 
the  same  mean  kinetic  energy,  or 

l/2miWi2  =  l/2m2it22.  (8) 

Dividing  equation  (7)  by  equation  (8),  we  have 

77-1   =   7i2, 

or  under  the  same  conditions  of  temperature  and  pressure  equal 
volumes  of  the  two  gases  contain  the  same  number  of  molecules. 
This  is  the  hypothesis  of  Avogadro. 

Law  of  Graham.     If  the  fundamental  kinetic  equation  be  solved 
for  u,  we  have 

u  = 


but  v/mn  =  1/d,  where  d  is  the  density  of  the  gas,  and  therefore 

we  may  write 

AT7T 

(9) 

If  the  pressure  and  temperature  remain  constant,  it  is  evident  that 
the  mean  velocities  of  the  molecules  of  two  gases  are  inversely 
proportional  to  the  square  roots  of  their  densities,  a  law  which 
was  first  enunciated  by  Graham  as  the  result  of  his  experiments  on 
gaseous  effusion. 

Effusion  of  Gases.  Let  u\  and  U2  denote  the  mean  velocities  of 
translation  of  the  molecules  of  any  two  gases  whose  densities  are 
di  and  da  respectively.  Then,  if  both  pressure  and  temperature 
are  constant,  we  may  write,  according  to  equation  (9), 


GASES  31 

or,  letting  MI  and  M2  denote  the  molecular  weights  of  the  two 
gases,  we  have 

11.       r^L       /fl/r 

(10) 

The  tendency  of  gases  to  effuse  through  small  apertures  is  a  man- 
ifestation of  the  motion  of  their  molecules,  and  it  was  from  experi- 
ments on  the  rate  of  such  gaseous  effusion  that  Graham  was  led 
to  the  discovery  of  the  law  which  bears  his  name. 

By  observing  the  times  of  effusion,  at  constant  temperature,  of 
equal  volumes  of  two  different  gases  through  the  same  aperture, 
and  under  the  same  difference  of  pressure,  it  is  possible  to  deter- 
mine the  density  of  one  gas,  provided  that  of  the  other  is  known. 
Since  the  times  of  effusion,  ti  and  k,  are  inversely  proportional  to 
the  velocities,  Ui  and  uz}  equation  (10)  becomes 

ti 
t2 

This  method  of  determining  the  density  and  molecular  weight  of 
gases  was  developed  by  Bunsen,  and  has  proved  to  be  of  great 
value,  especially  in  cases  where  only  a  small  volume  of  gas  is 
available.  Thus,  Debierne*  applied  the  method  of  effusion  to  the 
determination  of  the  molecular  weight  of  the  radium  emanation 
and,  notwithstanding  the  very  small  volume  of  gas  obtainable, 
he  found  the  value  of  the  molecular  weight  to  be  220  which  is  in 
remarkably  close  agreement  with  222,  the  accepted  value. 

Mean  Velocity  of  Translation  of  a  Gaseous  Molecule.  By 
substituting  appropriate  values  for  the  various  magnitudes  in  the 
equation 


it  is  possible  to  calculate  the  mean  velocity  of  the  molecules  of 
any  gas.     Thus,  for  the  gram-molecule  of  hydrogen  at  0°  and  76 
cm.  pressure,  p  =  76  X  13.59  =  1033.3  gr.  per  sq.  cm.  =  1033.3 
X  980.6  dynes  per  sq.  cm.,  v  =  22,400  cc.,  and  mn  =  2.016  gr. 
Substituting  these  values  in  the  above  equation  we  have, 


4  /3  X  1033.3  X  980.6  X  22,400  __. 

u  =  y  -  2Q16  -  =  183,780  cm.  per  sec. 

*  Compt.  rend.  150,  1740  (1910). 


32  THEORETICAL   CHEMISTRY 

Thus  at  0°  the  molecule  of  hydrogen  moves  with  a  speed  slightly 
greater  than  one  mile  per  second.  This  enormous  speed  is  only 
attained  along  the  mean  free  path,  the  frequent  collisions  with 
other  molecules  rendering  the  actual  speed  much  less  than  that 
calculated. 

Equation  of  van  der  Waals.  As  has  been  pointed  out  in  a 
previous  paragraph,  the  gas  laws  are  merely  limiting  laws  and 
while  they  hold  quite  well  up  to  pressures  of  about  2  atmospheres, 
above  this  pressure  the  differences  between  the  observed  and  cal- 
culated values  become  steadily  larger.  In  the  case  of  hydrogen, 
Natterer  was  the  first  to  show  that  the  product  of  pressure  and 
volume  is  invariably  higher  than  it  should  be.  A  possible  explana- 
tion of  this  departure  from  the  gas  laws  was  offered  by  Budde, 
who  proposed  that  the  volume,  v,  in  the  equation  pv  =  RT,  should 
be  corrected  for  the  volume  occupied  by  the  molecules  them- 
selves. If  this  volume  correction  be  denoted  by  6,  then  the  gas 
equation  becomes 

p  (v  -  b)  =  RT,  (12) 

where  b  is  a  constant  for  each  gas.  Budde  calculated  the  value 
of  b  for  hydrogen  and  found  it  to  remain  constant  for  pressures 
varying  from  1000  to  2800  meters  of  mercury. 

While  Budde's  modification  of  the  gas  equation  is  quite  satis- 
factory in  the  case  of  hydrogen,  it  fails  when  applied  to  other  gases. 
In  general,  the  compressibility  at  low  pressures  is  considerably 
greater  than  can  be  accounted  for  by  Boyle's  law.  The  compressi- 
bility reaches  a  minimum  value,  and  then  increases  rapidly,  so  that 
pv  passes  through  the  value  required  by  the  law.  This  suggests 
that  there  is  some  other  correction  to  be  applied  in  addition  to 
the  volume  correction  introduced  into  the  gas  equation  by  Budde. 
In  1879,  van  der  Waals  pointed  out  that  in  the  deduction  of  Boyle's 
law  by  means  of  the  fundamental  kinetic  equation,  the  tacit 
assumption  is  made  that  the  molecules  exert  no  mutual  attraction. 
While  this  assumption  is  undoubtedly  justifiable  when  the  gas  is 
subjected  to  a  very  low  pressure,  it  no  longer  remains  so  when 
the  gas  is  strongly  compressed.  A  little  consideration  will  make 
it  apparent  that  when  increased  pressure  is  applied  to  a  gas,  the 
resulting  volume  will  become  less  than  that  calculated,  owing  to 
molecular  attraction.  In  other  words  the  molecular  attraction 
and  the  applied  pressure  act  in  the  same  direction  and  the  gas 


GASES  33 

behaves  as  if  it  were  subjected  to  a  pressure  greater  than  that 
actually  applied.  It  was  shown  by  van  der  Waals  that  this  cor- 
rection is  inversely  proportional  to  the  square  of  the  volume,  and 
since  it  augments  the  applied  pressure,  the  expression,  p  -f  a/v2, 
should  be  substituted  for  p  in  the  gas  equation,  a  being  the  con- 
stant of  molecular  attraction.  The  corrected  equation  then  be- 
comes -T 

(p  +  a/v2)  (v  -  b)  =  RT.   (  (13) 

— T 
This  is  known  as  the  equation  of  van  der  Waals.     It  is  applicable 

not  only  to  strongly  compressed  gases,  but  also  to  liquids  as  well. 
While  it  will  be  given  detailed  consideration  in  a  subsequent 
chapter,  it  may  be  of  interest  to  point  out  at  this  time,  the  satis- 
factory explanation  which  it  offers  of  the  experimental  results  of 
Amagat,  to  which  we  have  already  made  reference,  (page  24). 
When  v  is  large,  both  6  and  a/v2  become  negligible,  and  van  der 
Waals'  equation  reduces  to  the  simple  gas  equation,  pv  =  RT. 
We  may  predict,  therefore,  that  any  influence  tending  to  increase 
v  will  cause  the  gas  to  approach  more  nearly  to  the  ideal  condi- 
tion. This  is  in  accord  with  the  results  of  Amagat's  experiments, 
which  show  that  an  increase  of  temperature  at  constant  pressure, 
or  a  diminution  of  pressure  at  constant  temperature,  causes  the 
gas  to  tend  to  follow  the  simple  gas  laws.  The  equation  also 
offers  a  satisfactory  explanation  of  the  exceptional  behavior  of 
hydrogen  when  it  is  subjected  to  pressure.  As  we  have  seen,  pv 
for  all  gases,  except  hydrogen,  diminishes  at  first  with  increasing 
pressure,  reaches  a  minimum  value,  and  then  increases  regularly. 
Since  the  volume  correction  in  van  der  Waals'  equation  acts  in 
opposition  to  the  attraction  correction,  it  is  apparent  that  at 
low  pressures  the  effect  of  attraction  preponderates,  while  at 
high  pressures  the  volume  correction  is  relatively  of  more  import- 
ance. At  some  intermediate  pressure  the  two  corrections  counter- 
balance each^^ther,  and  LLjs  at  ±his_  point .. that  the  ga&  follows 
Boyle's  law  strictly^-  The  exceptional  behavior  of  hydrogen  may 
be  accounted  for  by  making  the  very  plausible  assumption  that 
the  attraction  correction  is  negligible  at  all  pressures  in  compari- 
son with  the  volume  correction. 

Vapor  Density  and  Molecular  Weight.  As  has  been  pointed  out 
in  an  earlier  chapter,  when  a  substance  can  be  obtained  in  the  gas- 
eous state,  the  determination  of  its  molecular  weight  resolves  itself 


34  THEORETICAL  CHEMISTRY 

into  finding  the  mass  of  that  volume  of  vapor  which  will  occupy 
22.4  liters  at  0°  and  76  cm.  It  is  inconvenient  to  weigh  a  volume 
of  gas  or  vapor  under  standard  conditions  of  temperature  and 
pressure,  but  by  means  of  the  gas  laws  the  determination  made 
at  any  temperature  and  under  any  pressure  can  be  reduced  to 
standard  conditions.  For  example,  suppose  v  cc.  of  gas  are  found 
to  weigh  w  grams  at  t°  and  p  cm.  pressure,  then  the  weight  in 
grams  of  22.4  liters  or  22,400  cc.  at  0°  and  76  cm.  will  be  given  by 
the  following  proportion,  in  which  M  denotes  the  molecular  weight 
of  the  substance  :  — 

pv  .    76  X  22,400 

--^ 


or 

M  =  w  X  76  X  22,400  X  (t  +  273) 
273  pv 

The  determination  of  vapor  density  may  be  effected  in  either  of 
two  ways;  (1)  we  may  determine  the  mass  of  a  known  volume  of 
vapor  under  definite  conditions  of  temperature  and  pressure,  or 
(2)  we  may  determine  the  volume  of  a  known  mass  under  definite 
conditions  of  temperature  and  pressure.  There  are  a  variety  of 
methods  for  the  determination  of  vapor  density;  but  for  our  pur- 
pose it  will  be  necessary  to  describe  but  two  of  them.  In  the 
method  of  Regnault  the  mass  of  a  definite  volume  of  vapor  is 
determined,  while  in  the  method  due  to  Victor  Meyer  we  measure 
the  volume  of  a  known  mass. 

Method  of  Regnault.  In  this  method  which  is  especially  adapted 
to  permanent  gases,  use  is  made  of  two  spherical  glass  bulbs  of 
approximately  the  same  capacity,  each  bulb  being  provided 
with  a  well-ground  stop-cock.  By  means  of  an  air  pump  one 
bulb  is  evacuated  as  completely  as  possible,  and  is  then  filled, 
at  definite  temperature  and  pressure,  with  the  gas  whose  density 
is  to  be  determined.  The  stop-cock  is  then  closed  and  the  bulb 
weighed,  the  second  bulb  being  used  as  a  counterpoise.  The  use 
of  the  second  bulb  is  largely  to  avoid  the  troublesome  corrections 
for  air  displacement  and  for  moisture,  each  bulb  being  affected  in 
the  same  way  and  to  nearly  the  same  extent.  The  volume  of 
the  bulb  may  be  obtained  by  weighing  it  first  evacuated,  and  then 
filled  with  distilled  water  at  known  temperature.  From  these 
results  we  may  calculate  the  mass  per  unit  of  volume;  or  we  may 


GASES 


35 


substitute  the  values  of  w,  v,  p  and  t  in  the  above  formula  and 
calculate  M,  the  molecular  weight.  This  method  was  used  by 
Morley*  in  his  epoch-making  research  on  the  densities  of  hydro- 
gen and  oxygen. 

Method  of  Victor  Meyer.     In  the  method  of  Victor  Meyer,  a 
weighed  amount  of  the  substance  is  vaporized,  and  the  volume 


Fig.  7 


which  it  would  have  occupied  at  the  temperature  of  the  room  and 
under  existing  barometric  pressure  is  determined.  The  apparatus 
of  Meyer,  shown  in  Fig.  7,  consists  of  an  inner  glass  tube  A,  about 
1  cm.  in  diameter  and  75  cm.  in  length.  This  tube  is  expanded 
into  a  bulb  at  the  lower  end,  while  at  the  top  it  is  slightly  enlarged 
and  is  furnished  with  two  side  tubes  C  and  E.  The  tube  A  is 
suspended  inside  a  heating  jacket  B,  containing  some  liquid 
the  boiling  point  of  which  is  about  20°  higher  than  the  vaporizing 
temperature  of  the  substance  whose  vapor  density  is  to  be  de- 

*  Smithsonian  Contributions  to  Knowledge,  (1895). 


36  THEORETICAL  CHEMISTRY 

termined.  The  side  tube  E  dips  beneath  the  surface  of  water  in  a 
pneumatic  trough  G,  and  serves  to  convey  the  air  displaced  from 
A  to  the  eudiometer  F.  By  means  of  the  side  tube  C,  and  the 
glass-rod  D,  the  small  bulb  containing  the  substance  can  be 
dropped  to  the  bottom  of  A.  To  carry  out  a  determination  of 
vapor  density  with  this  apparatus,  the  liquid  in  B  is  heated  to 
boiling  and  the  sealed  bulb  V,  containing  a  weighed  amount  of  the 
substance,  is  placed  in  position  on  the  rod  D,  the  corks  being 
tightly  inserted.  When  bubbles  of  air  cease  to  issue  from  E  in 
the  pneumatic  trough,  showing  that  the  temperature  within 
A  is  constant,  the  eudiometer  F}  full  of  water,  is  placed  over  the 
mouth  of  E,  and  the  bulb  V  is  allowed  to  drop  by  drawing  aside 
the  rod  D.  Air  bubbles  immediately  begin  to  issue  from  E  and  to 
collect  in  the  eudiometer.  When  the  air  ceases  to  collect,  the 
eudiometer  is  closed  by  the  thumb  and  is  removed  to  a  large  cyl- 
inder of  water  where  it  is  allowed  to  stand  long  enough  to  acquire 
the  temperature  of  the  room.  It  is  then  raised  or  lowered  until 
the  level  of  water  inside  and  outside  is  the  same,  when  the  volume 
of  air  is  carefully  read  off.  In  this  method,  the  substance  on 
vaporizing  displaces  an  equal  volume  of  air  which  is  collected 
and  measured,  this  observed  volume  being  that  which  the  vapor 
would  occupy  after  reduction  to  the  conditions  under  which  the 
air  is  measured.  It  is  evident  that  in  this  method  we  do  not 
require  a  knowledge  of  the  temperature  at  which  the  substance 
vaporizes.  Since  the  air  is  measured  over  water,  the  pressure  to 
which  it  is  subjected  is  that  of  the  atmosphere  diminished  by  the 
vapor  pressure  of  water  at  the  temperature  of  the  experiment. 
The  method  of  calculating  molecular  weights  from  the  observa- 
tions recorded  may  be  illustrated  by  the  following  example  :  — 
0.1  gram  of  benzene  (C6H6)  was  weighed  out,  and  when  vaporized, 
32  cc.  of  air  were  collected  over  water  at  17°  and  750  mm.  pressure. 
The  vapor  pressure  of  water  at  17°  is  14.4  mm.,  and  the  actual 
pressure  exerted  by  the  gas  is  750  —  14.4  =  735.6  mm.  Sub- 
stituting in  the  proportion 

pv  760  X  22,400 

''  - 


and  solving  for  M  we  have 

M  _  0.1  X  760  X  22,400  X  (17  +  273) 
M  ''  273  X  735.6  X  32 


GASES  37 

The  result  agrees  fairly  well  with  the  molecular  weight  of  ben- 
zene (78.05)  calculated  from  the  formula. 

Unless  a  vapor  follows  the  gas  laws  very  closely,  the  value  of  the 
molecular  weight  obtained  by  the  method  of  Victor  Meyer  will  be 
only  approximate,  but  this  approximate  value  will  be  sufficiently 
near  to  the  true  molecular  weight  to  enable  us  to  choose  between 
the  simple  formula  weight,  given  by  chemical  analysis,  and  some 
multiple  of  it. 

Results  of  Vapor-Density  Determinations.  As  the  result  of 
numerous  vapor-density  determinations  extending  over  a  wide 
range  of  temperatures,  much  important  data  has  been  collected 
concerning  the  number  of  atoms  contained  in  the  molecules  of  a 
large  number  of  chemical  compounds.  The  molecular  weights 
;  of  most  of  the  elementary  gases  are  double  their  atomic  weights, 
showing  that  their  molecules  are  diatomic.  In  like  manner  the 
molecular  weights  of  mercury,  zinc,  cadmium  and,  in  fact,  all  of 
the  vaporizable  metallic  elements  have  been  found  to  be  identi- 
cal with  their  atomic  weights.  The  molecules  of  sulphur, 
arsenic,  phosphorous  and  iodine  are  polyatomic,  if  they  are  not 
heated  to  too  high  a  temperature.  The  investigations  of  Meyer 
and  others  have  shown  that  the  vapor  densities  of  a  large  number 
of  substances  diminish  as  the  temperature  is  increased.  In  other 
words  as  the  temperature  is  raised  the  number  of  atoms  contained 
in  the  molecules  decreases.  The  molecular  weight  of  sulphur,  cal- 
culated from  its  vapor  density  at  temperatures  below  500°,  corre- 
sponds to  the  formula  S%.  If  the  vapor  of  sulphur  is  heated  to 
1100°,  the  molecular  weight  corresponds  to  the  formula  82.  In 
fact,  sulphur  in  the  form  of  vapor  may  be  represented  by  the  for- 
mulas Ss,  *S4,  /S>2,  or  even  S  according  to  the  temperature  at  which 
its  vapor  density  is  determined.  Iodine  behaves  similarly,  the 
molecules  being  diatomic  between  200°  and  600°,  while  at  temper- 
atures above  1400°  the  vapor  density  has  about  one-half  its  value 
at  the  lower  temperature,  showing  a  complete  breaking  down  of 
the  diatomic  molecules  into  single  atoms.  Heating  to  yet  higher 
temperatures  has  failed  to  reveal  any  further  decomposition. 
This  phenomenon  is  not  confined  to  the  molecules  of  the  elements 
alone,  but  is  also  met  with  in  the  case  of  the  molecules  of  chemical 
compounds.  The  vapor  density  of  arsenious  oxide  between  500° 
and  700°  corresponds  to  the  formula  As4Oe.  As  the  temperature 
is  raised,  the  vapor  density  becomes  steadily  smaller  until,  at 


38  THEORETICAL  CHEMISTRY 

1800°,  the  calculated  molecular  weight  corresponds  to  the  for- 
mula As203.  In  like  manner  ferric  and  aluminium  chlorides  have 
been  shown  to  have  molecular  weights  at  low  temperatures  corre- 
sponding to  the  formulas,  Fe2Cl6  and  A12C16.  The  commonly- 
used  formulas,  FeCl3  and  AlCla,  represent  their  molecular  weights 
at  high  temperatures  only.  The  experimental  difficulties  attend- 
ing vapor  density  determinations  increase  as  the  temperature  is 
raised,  owing  chiefly  to  the  deformation  of  the  apparatus  when  the 
material  of  which  it  is  constructed  approaches  its  melting-point. 
Glass  which  can  be  used  at  relatively  low  temperatures  only,  has 
been  replaced  by  specially  resistant  varieties  of  porcelain  which 
may  be  used  up  to  temperatures  of  1500°  or  1600°.  Platinum 
vessels  retain  their  shape  up  to  temperatures  between  1700° 
and  1800°.  Measurements  up  to  2000°  have  been  effected  by 
Nernst  and  his  pupils.*  In  their  experiments  use  was  made  of 
a  vessel  of  iridium,  the  inside  and  outside  of  which  was  surrounded 
with  a  cement  of  magnesia  and  magnesium  chloride,  the  entire 
apparatus  being  heated  electrically.  With  this  apparatus  they 
showed  that  the  molecular  weight  of  sulphur  between  1800°  and 
2000°  is  48,  indicating  that  the  diatomic  molecule  is  approxi- 
mately 50  per  cent  broken  down  into  single  atoms. 

Abnormal  Vapor  Densities.  In  all  of  the  cases  cited  above 
the  molecular  weight  calculated  from  the  vapor  density  corre- 
sponds either  with  the  simple  formula  weight,  as  determined  by 
chemical  analysis,  or  with  a  multiple  thereof.  In  no  case  is  there 
any  evidence  of  a  breaking  down  of  the  simple  molecule  into  its 
constituents.  Substances  are  known,  however,  the  molecular 
weights  of  which,  calculated  from  their  vapor  densities,  are  less 
than  the  sum  of  the  atomic  weights  of  their  constituents.  For 
example,  the  vapor  density  of  ammonium  chloride  was  found  to 
be  0.89,  while  that  corresponding  to  the  formula  NH4C1  should  be 
1.89.  Similar  results  have  been  obtained  with  phosphorus  penta- 
chloride,  nitrogen  peroxide,  chloral  hydrate  and  numerous  other 
substances.  The  phenomenon  can  be  explained  in  either  of  the 
two  following  ways:  (1)  that  the  molecule  has  undergone  a  com- 
plete disruption,  or  (2)  that  the  substance  does  not  follow  the  law 
of  Avogadro.  Until  the  former  explanation  was  shown  to  be  cor- 
rect; the  latter  was  accepted  and  for  a  time  the  law  of  Avogadro 

*  Wartenberg.  Zeit.  anorg.  Chem.,  56,  320  (1907). 


GASES 


39 


fell  into  disrepute.  In  1857,  Deville  showed  that  numerous  chem- 
ical compounds  are  broken  down  or  "  dissociated  "  at  high  tem- 
peratures. Shortly  afterward  Kopp  suggested  that  the  abnormal 
vapor  densities  of  such  substances  as  ammonium  chloride,  phos- 
phorus pentachloride,  etc.,  might  be  due  to  thermal  dissociation. 
If  ammonium  chloride  underwent  complete  dissociation,  one  mole- 
cule of  the  salt  would  yield  one  molecule  of  ammonia  and  one 
molecule  of  hydrochloric  acid  gas,  and  the  vapor  density  of  the 
resulting  mixture  would  be  one-half  of  that  of  the  undissociated 
substance,  a  deduction  in  complete  agreement  with  the  results  of 
experiment.  It  remained  to  prove  that  the  products  of  this  sup- 
posed dissociation  were  actually  present. 


Hydrogen 


H j drogen 


Fig.  8 


The  first  to  offer  an  experimental  demonstration  of  the  simul- 
taneous formation  of  ammonia  and  hydrochloric  acid,  when  am- 
monium chloride  is  heated,  was  Pebal.*  The  apparatus  which  he 
devised  for  this  purpose  is  shown  in  Fig.  8.  It  consisted  of  two 
tubes  T  and  t,  the  latter  being  placed  within  the  former  as  indi- 


Lieb.  Ann.,  123,  199  (1862). 


40 


THEORETICAL  CHEMISTRY 


cated  in  the  sketch.  Near  the  top  of  the  inner  tube,  which  was 
drawn  down  to  a  smaller  diameter,  was  a  porous  plug  of  asbestos, 
C,  upon  which  was  placed  a  little  ammonium  chloride.  A  stream 
of  dry  hydrogen  was  passed  through  the  apparatus  by  means  of 
the  tubes  A  and  B,  the  former  entering  the  outer  tube  and  the 
latter  the  inner  tube.  The  entire  apparatus  was  heated  to  a  tem- 
perature above  that  necessary  to  vaporize  the  ammonium  chlor- 
ide. If  the  salt  undergoes  dissociation  into  ammonia  and  hydro- 
chloric acid,  the  former  being  less  dense  than  the  latter,  would 
diffuse  more  rapidly  through  the  plug  C  and  the  vapor  below  the 
plug  would  be  relatively  richer  in  ammonia  than  the  vapor  above 
it.  The  current  of  hydrogen  through  B  would  therefore  sweep 
out  from  the  lower  part  of  t  an  excess  of  ammonia,  while  the  cur- 
rent through  A  would  carry  out  from  T  an  excess  of  hydrochloric 


Nitrogen 


Fig.  9 

acid.  By  holding  strips  of  moistened  litmus  paper  in  the  currents 
of  gas  issuing  from  E  and  F,  it  was  possible  for  Pebal  to  test  the 
correctness  of  Kopp's  idea.  He  found  that  the  gas  issuing  from 
E  had  an  acid  reaction  while  that  escaping  from  F  had  an  alkaline 
reaction.  It  would  at  first  sight  appear  that  Pebal  had  demon- 
strated beyond  question  that  ammonium  chloride  undergoes  dis- 
sociation into  ammonia  and  hydrochloric  acid. 

It  was  pointed  out,  however,  that  Pebal  had  heated  the  ammo- 
nium chloride  in  contact  with  a  foreign  substance,  asbestos,  and 


GASES  41 

that  this  might  have  acted  as  a  catalyst,  promoting  the  decomposi- 
tion into  ammonia  and  hydrochloric  acid.-  This  objection  was 
removed  by  the  ingenious  experiment  of  Than.*  He  devised  a 
modification  of  PebaPs  apparatus,  as  shown  in  Fig.  9.  In  the 
horizontal  tube,  A  B,  the  ammonium  chloride  was  placed  at  F  and 
a  porous  plug  of  compressed  ammonium  chloride  was  introduced 
at  G.  The  tube  was  heated  and  nitrogen  passed  in  at  C.  The 
reactions  of  the  currents  of  gas  issuing  at  D  and  E  were  tested 
with  litmus,  as  in  Pebal's  experiment,  and  it  was  found  that  the 
gas  escaping  from  D  was  alkaline,  while  that  issuing  from  E  was 
acid.  This  experiment  proved  beyond  question  that  the  vapor 
of  ammonium  chloride  is  thermally  dissociated  into  ammonia 
and  hydrochloric  acid.  Experiments  on  other  substances  whose 
vapor  densities  are  abnormally  small  show  that  a  similar  explan- 
ation is  applicable,  and  thus  furnish  a  confirmation  of  the  law  of 
Avogadro. 

Calculation  of  the  Degree  of  Dissociation.  Since  the  density 
of  a  dissociating  vapor  decreases  with  increase  in  temperature, 
it  is  important  to  be  able  to  calculate  the  degree  of  dissociation  at 
any  one  temperature.  This  is  clearly  equivalent  to  ascertaining 
the  extent  to  which  the  reaction 

NH4C1  <±  NH3  +  HC1 

has  proceeded  from  left  to  right.  This  can  be  determined  easily 
from  the  relation  of  vapor  density  to  dissociation.  If  we  start 
with  one  molecule  of  gas  and  let  a  represent  the  percentage  dis- 
sociation, then  1  —  a  will  denote  the  percentage  remaining  un- 
dissociated.  If  one  molecule  of  gas  yields  n  molecules  of  gaseous 
products,  the  total  number  of  molecules  present  at  any  time  will 

be 

(1  -  a)  +  na  =  1  +  (n  -I)  a. 

The  ratio,  1  :  1  +  (n  —  1)  a,  will  be  the  same  as  the  ratio  of  the 
density  dz  of  the  dissociated  gas  to  its  density  in  the  undissociated 
state  di,  or 

1:  1  +  (n-  I)  a  =  dt-.dii 


solving  this  proportion  for  a,  we  have 

di  —  c?2 


(n  -  1)  dz 
*  Lieb.  Ann.,  131,  129  (1864), 


(14) 


42 


THEORETICAL  CHEMISTRY 


The  vapor  density  of  nitrogen  peroxide  has  been  measured  by 
E.  and  L.  Natanson,*  and  the  degree  of  dissociation  at  the  differ- 
ent temperatures  calculated  by  means  of  the  preceding  formula. 
The  following  table  gives  their  results. 

DISSOCIATION  OF  NITROGEN  PEROXIDE,   N2O4 

ATMOSPHERIC  PRESSURE 
(Density  of  N2O4  =  3.18;   of  NO2  +  NO2  =  1.59;   of  air  =  1.00) 


Temperature, 
(decrees) 

Density  of  Gas. 

Percentage  Dis- 
sociation. 

26.7 

2.65 

19.96 

35.4 

2.53 

25.65 

39.8 

2.46 

29.23 

49.6 

2.27 

40.04 

60.2 

2.08 

52.84 

70.0 

1.92 

65.57 

80.6 

.80 

76.61 

90.0 

.72 

84.83 

100.1 

.68 

89.23 

111.3 

.65 

92.67 

121.5 

.62 

96.23 

135.0 

.60 

98.69 

154.0 

.58 

100.00 

It  will  be  observed  that  the  dissociation  of  nitrogen  peroxide 
is,  at  first  nearly  proportional  to  the  temperature,  and  then 
increases  more  rapidly  until,  when  about  four-fifths  of  the  mole- 
cules of  N2O4  are  broken  down,  the  dissociation  proceeds  slowly 
to  completion. 

Specific  Heat.  The  addition  of  heat  energy  to  a  body  causes 
its  temperature  to  rise.  The  ratio  of  the  amount  of  heat  supplied 
to  the  resulting  rise  in  temperature  is  called  the  heat  capacity  of 
the  body;  obviously  its  value  is  dependent  upon  the  initial  temper- 
ature of  the  body.  The  specific  heat  of  a  substance  may  be  defined 
as  the  heat  capacity  of  unit  mass  of  the  substance.  If  dt  represents 
the  increment  of  temperature  due  to  the  addition  of  dQ  units  of 
heat  energy  to  m  grams  of  any  substance,  then  its  specific  heat,  c, 
will  be  given  by  the  equation 


m     dt 
*  Wied.  Ann.,  24,  454  (1885);  27,  606  (1886). 


(15) 


GASES  43 

Specific  Heat  at  Constant  Pressure  and  Constant  Volume.     It 

is  well  known  that  the  specific  heat  of  a  gas  depends  upon  the 
conditions  under  which  it  is  determined.  If  a  definite  mass  of 
gas  is  heated  under  constant  pressure,  the  value  of  the  specific 
heat,  cp,  is  different  from  the  value  of  the  specific  heat,  cv,  ob- 
tained when  the  pressure  varies  and  the  volume  remains  con- 
stant. The  value  of  CP  is  invariably  greater  than  that  of 
d,.  When  heat  is  supplied  to  a  gas  at  constant  pressure  not  only 
does  its  temperature  rise,  but  it  also  expands,  and  thus  does  ex- 
ternal work.  On  the  other  hand,  if  the  gas  be  heated  in  such  a 
way  that  its  volume  cannot  change,  none  of  the  heat  supplied 
will  be  used  in  doing  external  work,  and  consequently  its  heat 
capacity  will  be  less  than  when  it  is  heated  under  constant  pressure. 
The  recognition  by  Mayer,  in  1841,  of  the  cause  of  this  difference 
between  the  two  specific  heats  of  a  gas,  led  him  to  his  celebrated 
calculation  of  the  mechanical  equivalent  of  heat,  and  the  enun- 
ciation of  the  first  law  of  thermodynamics.  Mayer  observed  that 
the  difference  between  the  quantity  of  heat  necessary  to  raise  the 
temperature  of  1  gram  of  air  1°  C.  at  constant  pressure,  and  at 
constant  volume  respectively,  was  0.0692  calorie,  or 

cv  -  CD  =  0.0692  cal. 

That  is  to  say,  0.0692  calorie  is  the  amount  of  heat  energy  which 
is  equivalent  to  the  work  required  to  expand  1  gram  of  air  1/273 
of  its  volume  at  0°.  Imagine  1  gram  of  air  at  0°  enclosed  within 
a  cylinder  having  a  cross-section  of  one  square  centimeter,  and 
furnished  with  a  movable,  frictionless  piston.  Since  1  gram  of 
air  under  standard  conditions  of  temperature  and  pressure  occu- 
pies 773.3  cc.,  the  distance  between  the  piston  and  the  bottom  of 
the  cylinder  will  be  773.3  cm.  If  the  temperature  be  raised  from 
0°  to  1°,  the  piston  will  rise  1/273  X  773.3  =  2.83  cm.,  and  since 
the  pressure  of  the  atmosphere  is  1033.3  grams  per  square  centi- 
meter, the  external  work  done  by  the  expanding  gas  will  be 

1033.3  X  2.83  =  2924.3  gm.  cm. 

This  is  evidently  equivalent  to  0.0692  calorie  and  therefore,  the 
equivalent  of  1  calorie  in  mechanical  units,  J,  will  be 


44  THEORETICAL  CHEMISTRY 

a  value  agreeing  very  well  with  the  best  recent  determinations  of 
the  mechanical  equivalent  of  heat. 

The  difference  between  the  two  specific  heats  may  be  easily 
calculated  in  calories  from  the  fundamental  gas  equation.  Start- 
ing with  1  mol.  of  gas,  and  remembering  that  when  a  gas  expands 
at_constant  pressure,  the  product  of  pressure  and  change  in  volume 
is  a  measure  of  the  work  done,  we  have,  at  temperature  7\, 

pvi  =  RT1} 

where  Vi  is  the  molecular  volume.  Raising  the  temperature  to 
T2,  the  corresponding  molecular  volume  being  v2,  we  have  for 
the  work  done  during  expansion 

p(V2  -Vl)  =  R(T2-  Ti). 
If  Tz  —  TI  =  1°,  then  the  equation  reduces  to 

p  (vz  -  *>i)  =  R. 

Since  the  difference  between  the  molecular  heats*  at  constant 
pressure  and  Constant  volume  is  equivalent  to  the  external  work 
involved  when  the  temperature  of  1  mol.  of  gas  is  raised  1°,  we 
have 

M  (cp  -  cv)  =  p(v2  -  v^, 

where  M  is  the  molecular  weight  of  the  gas;  and  therefore 

M  (cp  -  cv)  =  R  =  2  calories.  (16) 

In  words,  the  difference  of  the  molecular  heats  of  any  gas  at 
constant  pressure  and  at  constant  volume  is  2  calories.  The 
specific  heat  of  a  gas  at  constant  pressure  can  be  readily  deter- 
mined, by  passing  a  definite  volume  of  the  gas,  heated  under  con- 
stant pressure  to  a  known  temperature,  through  the  worm  of  a 
calorimeter  at  such  a  rate  that  a  constant  difference  is  maintained 
between  the  temperature  of  the  entering  and  the  temperature  of 
the  escaping  gas.  Thus  the  number  of  calories  which  causes  a 
definite  thermal  change  in  a  certain  volume  of  the  gas  is  deter- 
mined, and  from  this  it  is  an  easy  matter  to  calculate  the  specific 
heat,  cp. 

*  The  molecular  heat  of  a  gas  is  equal  to  the  product  of  its  specific  heat  and  its 
molecular  weight. 


GASES 


45 


The  molecular  heat  at  constant  pressure  for  all  gases  ap- 
proaches the  limiting  value,  6.5,  at  the  absolute  zero.  This 
relation,  due  to  Le  Chatelier,  may  be  expressed  thus, 

Mcp  =  6.5  +  aT,  (17) 

where  a  is  a  constant  for  each  gas.  The  value  of  a  for  hydrogen, 
oxygen,  nitrogen  and  carbon  monoxide  is  0.001,  for  ammonia, 
0.0071  and  for  carbon  dioxide,  0.0084.  As  the  complexity  of  the 
gas  increases  the  value  of  a  becomes  numerically  greater. 

The  experimental  determination  of  the  specific  heat  of  a  gas  at 
constant  volume  is  difficult  and  the  results  obtained  are  not 
trustworthy.  The  chief  cause  of  the  inaccuracy  of  the  results 
is  that  the  vessel  containing  the  gas  absorbs  so  much  more  heat 
than  the  gas  itself  that  the  correction  is  many  times  larger  than 
the  quantity  to  be  measured.  The  specific  heat  at  constant  vol- 
ume is  almost  always  obtained  by  indirect  methods,  as  for  example 
by  means  of  the  preceding  formula, 

M  (cp  -  cv)  =  R  =  2  cal., 

in  wnich  the  values  of  M  and  cp  are  known. 

The  molecular  heats  of  some  of  the  commoner  gases  and  vapors 
are  given  in  the  subjoined  table  together  with  the  ratio  CP/CV. 


MOLECULAR  SPECIFIC  HEATS 


Gas. 

Mcp 

Mcv 

cp/cv=y 

Argon  . 

1  66 

Helium  

1  66 

Mercury  

1  66 

Hydrogen.  

6  88 

4  88 

41 

Oxygen 

6  96 

4  96 

40 

Nitrogen 

6  93 

4  93 

41 

Chlorine 

8  58 

6  58 

30 

Bromine  . 

8  88 

6  88 

29 

Nitric  oxide  

6  95 

4  95 

40 

Carbon  monoxide  

6.86 

4.86 

.41 

Hydrochloric  acid  
Carbon  dioxide 

6.68 
9  55 

4.68 
7  55 

.43 
26 

Nitrous  oxide 

9  94 

7  94 

25 

Water  

8  65 

6  65 

28 

Sulphur  dioxide  

9.88 

7.88 

.25 

Ozone  

.29 

Ether  

35  51 

33  & 

06 

46  THEORETICAL  CHEMISTRY 

The  Ratio  of  the  Two  Specific  Heats.  There  are  two  methods 
by  which  the  ratio  cp/cv  can  be  determined  directly,  one  due  to 
Clement  and  Desormes*  and  the  other  due  to  Kundt.  f 

Method  of  Clement  and  Desormes.  The  gas  to  be  investigated 
is  introduced  into  a  large  glass  balloon  under  a  pressure  PI  which 
is  only  a  trifle  greater  than  atmospheric  pressure,  P.  The  flask 
is  now  opened  for  a  moment  in  order  to  allow  the  pressure  within 
the  vessel  to  fall  to  that  of  the  atmosphere,  after  which  it  is  again 
closed.  The  expansion  of  the  gas,  resulting  from  the  momentary 
opening  of  the  flask,  causes  a  slight  lowering  of  its  temperature. 
Thermal  equilibrium  is  again  restored  by  the  influx  of  heat  from 
without,  which  in  turn,  causes  the  pressure  in  the  vessel  to  increase 
to  a  value  P2  slightly  greater  than  atmospheric  pressure.  Let  PI  = 
P  -f:  pi  and  P2  =  P  +  PI,  where  pi  and  p2  are  small  in  compari- 
son with  P.  If  V  denotes  the  volume  of  the  flask,  the  volume 
of  gas  which  escapes  when  the  flask  is  opened  is  V(p\  —  pz)/P, 
and  the  work  done  against  the  atmosphere  is  V(p\  —  p%).  This 
is  the  amount  of  work  resulting  from  the  expansion  of  the 
gas,  which  is  assumed  to  take  place  so  rapidly  that  the  system 
neither  gains  nor  loses  heat.  The  pressure  of  the  gas  increases 
from  P  to  P  +  pz  in  consequence  of  the  restoration  of  thermal 
equilibrium  following  the  closing  of  the  flask,  and  the  value  of  t, 
the  number  of  degrees  by  which  the  gas  is  cooled  below  the  tem- 
perature of  the  room,  T,  can  be  calculated  by  means  of  the  pro- 
portion, 

t/T  =  pt/P. 

Since  the  influx  of  heat  occurs  at  constant  volume,  and  as  the 
number  of  gram-molecules  of  gas  contained  in  the  flask  is 

n  =  PV/RT, 

it  follows  that  the  number  of  calories  taken  up  by  the  gas  must  be 
PV 


This  quantity  of  heat  is  equivalent  to  V(p\  —  pz),  the  work  per- 
formed by  the  gas.     Therefore,  we  have 


*  Jour,  de  phys.,  89,  321,  428  (1819). 

t  Pogg.  Ann.,  128,  497  (1866);   135,  337,  527  (1868). 


GASES  47 

or 

rii   —  nn  = 


Pi-  P2  =  ~  Me,. 


But,  according  to  eq.  (16), 

R  =  M  (cv  -  cv). 
Hence 


Owing  to  the  fact  that  some  heat  ;s  removed  from  the  walls  of  the 
flask  during  the  escape  of  the  undercooled  gas,  the  observed  value 
t,  and  the  corresponding  value  of  p2,  will  be  too  small,  and  therefore 
the  calculated  value  of  7  will  likewise  be  too  small.  This  source 
of  error  can  be  minimized  by  using  a  large  flask  and  making  the 
differences  of  pressure  as  small  as  possible. 

Method  of  Kundt.     According  to  the  formula  of  Laplace  for  the 
velocity  of  transmission  of  a  sound  wave  in  a  gas,  we  have 

v  = 

in  which  p  and  d  denote  the  pressure  and  density  of  the  gas,  and 
7  is  the  ratio  of  the  two  specific  heats.  If  the  wave  velocities  in 
two  different  gases,  whose  densities  are  di  and  d2  under  the  Same 
conditions  of  temperature  and  pressure,  be  denoted  by  Vi  and  v2, 
we  may  write 


or  replacing  the  densities  of  these  gases  by  their  respective  molec- 
ular weights,  MI  and  M2,  we  have 


The  ratio  of  the  velocities  of  the  two  waves  can  be  measured  by 
means  of  the  apparatus  shown  in  Fig.  10.  A  wide  glass  tube 
about  1J  meters  in  length  is  furnished  with  two  side  tubes,  E  and 
F.  Into  one  end  of  the  tube  is  inserted  the  glass  rod  BD  which 
is  clamped  at  its  middle  point  by  a  tightly  fitting  cork,  C.  The 
other  end  of  the  tube  is  closed  by  means  of  the  plunger  A.  A 
small  amount  of  lycopodium  powder  is  placed  upon  the  bottom  of 


48  THEORETICAL  CHEMISTRY 

the  tube  and  is  distributed  uniformly  by  gently  tapping  the  walls 
of  the  tube.  The  gas  in  which  the  velocity  of  the  sound  wave 
is  to  be  determined,  is  introduced  into  the  tube  through  E,  and 
the  displaced  air  escapes  at  F.  When  the  tube  is  filled,  E  and  F 
are  closed  by  means  of  rubber  caps,  and  a  piece  of  moistened 
chamois  leather  is  drawn  along  BD  causing  it  to  vibrate  longitudi- 


>  "fi"  -'I  

y 

IT 

Fig.  10 

nally  and  to  emit  a  shrill  note.  The  vibrations  are  taken  up  by 
the  gas  in  the  tube  and  the  powder  arranges  itself  in  a  series  of 
heaps  corresponding  to  the  nodes  of  vibration.  If  the  nodes  are 
not  sharply  defined,  then  A  should  be  moved  in  or  out  until  they 
become  so.  If  Xi  is  the  distance  between  two  heaps  or  nodes, 
then  2  \i  will  be  the  wave  length  of  the  note  emitted  by  the  rod 
BD,  and  if  n  represents  the  number  of  vibrations  per  second 
of  the  note  emitted,  we  have  for  the  velocity  of  sound  in  the  gas 

Vi  =  2  n\i. 
Similarly  if  a  second  gas  be  introduced  into  the  tube  we  shall  have 

i>2  =  2  n\2. 
Therefore, 

H-  «» 

Substituting  in  equation  (19),  we  have 


or 


If  the  second  gas  is  air,  as  is  usually  the  case,  72  =  1.405  and  M2 
=  28.74,  (mol.  wt.  of  hydrogen  -f-  density  of  hydrogen  referred 
to  air,  or  2  -^  0.0696  =  28.74)  or  equation  (21)  becomes 

•  (22) 


GASES  49 

Thus,  7  for  any  gas  can  be  determined  by  this  method,  provided 
we  know  its  value  for  another  gas  of  known  molecular  weight. 

Specific  Heat  of  Gases  and  the  Kinetic  Theory.  In  terms  of 
the  kinetic  theory,  the  energy  of  a  gas  may  be  considered  to  be 
made  up  of  three  parts:  (1)  the  translational  energy  of  the  mole- 
cules, commonly  termed  their  kinetic  energy,  (2)  the  intramolec- 
ular kinetic  energy,  and  (3)  the  potential  energy  due  to  inter- 
atomic attraction  within  the  molecules.  When  a  gas  is  heated 
at  constant  volume,  all  three  of  these  factors  of  the  total  energy 
of  the  molecule  may  be  affected.  It  is  fair  to  assume,  however, 
that  when  a  monatomic  gas,  such  as  mercury  vapor,  is  heated, 
all  of  the  heat  energy  supplied  is  used  to  augment  the  translational 
kinetic  energy  of  the  molecules.  As  we  have  seen,  the  fundamen- 
tal kinetic  equation 

pv  =  1/3  nmu2 

may  be  written 

pv  =  2/3  -  1/2  nmu2, 

and  since  1/2  nmu2  represents  the  total  kinetic  energy  of  the  gas, 
we  have 

pv  =  2/3  kinetic  energy  of  1  mol, 

or 

kinetic  energy  of  1  mol  =  3/2  pv. 

But  pv  =  2  T  calories  therefore 

kinetic  energy  of  1  mol  =  3  T  cal. 

The  kinetic  energy  of  a  constant  volume  of  any  gas  at  the  temper- 
atures TI  and  T2,  is  given  by  the  following  equations- 

3/2  Plv  =  3  T7!,  (23) 

and 

3/2  pzv  =  3  Tz.  (24) 

Subtracting  (23)  from  (24)  we  obtain 

3/2(p2-Pl)v  =  3(T2-  TJ,  (25) 

and  for  an  increase  in  temperature  of  1°,  (25)  becomes 
3/2  (p2  -  pi)  v  =  3  cal. 


50  THEORETICAL  CHEMISTRY 

The  molecular  kinetic  energy  of  one  mol  of  a  monatomic  gas 
at  constant  volume  is  thus  increased  by  3  calories  for  each  degree 
rise  in  temperature.  As  has  already  been  shown, 

M  (cp  —  Cv)  =  2  cal., 
therefore,  since  Mcv  =  3  calories,  McP  =  3  +  2  =  5  calories,  and 

i-1-86-  (2G) 

This  value  of  7  is  in  perfect  agreement  with  the  results  of  the 
experiments  on  mercury  vapor  which  is  known  to  be  monatomic. 
The  converse  of  this  method  has  been  employed  by  Ramsay  to 
prove  that  the  rare  gases  of  the  atmosphere  are  monatomic,  the 
value  of  7  for  all  of  these  gases  being  1.66.  In  the  case  of  poly- 
atomic molecules  the  heat  energy  supplied  is  not  only  used  in 
increasing  their  translational  kinetic  energy,  but  also  in  the  per- 
formance of  work  within  the  molecule.  The  value  of  the 
internal  work  is  indeterminate,  but  it  is  without  doubt  constant 
for  any  one  gas.  If  the  internal  work  be  represented  by  a,  then 
the  value  of  the  ratio  of  the  two  specific  heats  will  be 


p  _  5  +  a 
7  ~  Mcv  ~  3  +  a  < 

Reference  to  the  table  on  p.  45,  giving  the  value  of  7  for  differ- 
ent gases,  will  show  that  this  deduction  from  the  kinetic  theory 
is  in  perfect  agreement  with  the  experimental  facts.  With 
increasing  complexity  of  the  molecule,  it  is  apparent  that  the 
amount  of  heat  expended  in  doing  internal  work  should  increase, 
and  therefore  the  specific  heat  should  increase  also.  Inspection 
of  the  tables  confirms  this  deduction.  The  specific  heat  of  mona- 
tomic gases  is  independent  of  the  temperature  while  the  specific 
heat  of  polyatomic  gases  increases  slightly.  These  results  may 
justly  be  regarded  as  among  the  greatest  triumphs  of  the  kinetic 
theory  of  gases. 

REFERENCE 

The  Kinetic  Theory  of  Gases.     Meyer.     Translated  by  R.  E.  Baynes. 


'GASES  51 

« 

PROBLEMS 

The  volume  of  a  quantity  of  gas  is  measured  when  the  barometer 
stands  at  72  cm.,  and  is  found  to  be  646  cc.:  what  would  its  volume  be 
>  >rmal  pressure?  Ans.  612  cc. 

Vt  what  pressure  would  the  gas  in  the  preceding  problem  have  a 
line  of  ."  0  cc.?  Ans.  80.19  cm. 

A  certain  quantity  of  oxygen  occupies  a  volume  of  300  cc.  at  0°: 
,  its  volume  ai  91°.  y  a  &  c  c~ 

3^3®  The  weight  of  a  liter  of  air  under  standard  conditions  is  1.293  grams: 
'to  whai  temperature  must  the  air  be  heated  so  that  it  may  weigh  exactly 
1  gram  per  liter?  J  3'J  * 

(5)  At  what  temperature  will  the  volume  of  a  given  mass  of  gas  be 
exactly  double  what  it  is  at  17°?     />       *«"•**  Ans.  307°. 

6.  On  heating  a  certain  quantity  of  mercuric  oxide  it  is  found  to  give 
off  380  cc.  of  oxygen,  the  temperature  being  23°  and  the  barometric  height 
74  cm. ;  what  would  be  the  volume  of  the  gas  under  standard  conditions? 

7.  A  liter  of  air  weighs  1.293  grams  under  standard  conditions.     At 
what  temperature  will  a  liter  of  air  weigh  1  gram,  the  pressure  being  72 
cm.?  Ans.  61.43° 

8.  A  quantity  of  air  at  atmospheric  pressure  and  at  a  temperature  of 
7°  is  compressed  until  its  volume  is  reduced  to  one-seventh,  the  temper- 
ature rising  20°  during  the  process:   find  the  pressure  at  the  end  of  the 
operation. 

9.  The  weight  of  a  liter  of  nitrogen  under  standard  conditions  is  1.2579 
grams.     Calculate  the  specific  gas  constant,  r.          Ans.  3007  gm.  cm. 

10.  The  time  of  outflow  of  a  gas  is. 21.4  minutes,  the  corresponding 
time  for  hydrogen  is  5.6  minutes.     Find  the  molecular  weight  of  the  gas. 

11.  The  times  of  effusion  of  equal  volumes  of  air  and  carbon  dioxide 
are  36.9  sec.,  and  45.3  sec.  respectively:    the  density  of  air  referred  to 
0  =  16  is  14.477.     Calculate  the  molecular  weight  of  carbon  dioxide. 

12.  Calculate  the  molecular  weight  of  chloroform,  from  the  following 
data : — 

Weight  of  chloroform  taken ....................     0.220  gr. 

Volume  of  air  collected  over  water ...............      45.0  cc. 

Temperature  of  air ...........................       20° 

Barometric  pressure ...........................     755.0  mm. 

Pressure  of  aqueous  vapor  at  20° ................       17.4  mm. 

Ans.  121.1 

13.  The  density  of  a  gas  is  0.23  referred  to  mercury  vapor.     What  is 
its  molecular  weight? 

14.  Phosphorus  pentachloride  dissociates  according  to  the  equation 

PC15  <=>  PC13  +  C12. 


52  THEORETICAL  CHEMISTRY 

The  molecular  weight  PC15  is  208.28.    At  182°  the  density  is  73.5  and 
at  203°  it  is  62.    Find  the  degree  of  dissociation  at  the  two  temperatures. 

Ans.  «i82°  =  0.417,  a230o  =  0.68. 

15.  The  specific  heat  at  constant  volume  for  argon  is  0.075,  and  its 
molecular  weight  is  40.    How  many  atoms  are  there  in  the  molecule? 

16.  What  is  the  specific  heat  of  carbon  dioxide  at  constant  volume,  its 
molecular  weight  being  44  and  the  temperature  50°. 

17.  The  specific  heat  for  constant  pressure  of  benzene  is  0.295:  what  is 
the  specific  heat  for  constant  volume? 


CHAPTER  II 
LIQUIDS 

General  Characteristics  of  Liquids.  The  most  marked  char- 
acteristic of  the  liquid  state  is  that  a  given  mass  of  liquid  has  a 
definite  volume  but  no  definite  form.  The  volume  of  a  liquid  is 
dependent  upon  temperature  and  pressure  but  to  a  much  smaller 
degree  than  is  the  volume  of  a  gas.  The  formulas  in  which  the  vol- 
ume of  a  liquid  is  expressed  as  a  function  of  temperature  and  pres- 
sure are  largely  empirical,  and  contain  constants  dependent  upon 
the  nature  of  the  liquid.  This  is  undoubtedly  due  to  the  fact  that 
in  the  liquid  state  the  molecules  are  much  less  mobile  than  in  the 
gaseous  state.  The  distance  between  contiguous  molecules  being 
much  less  in  liquids  than  in  gases,  the  mutual  attraction  is  in- 
creased while  the  mobility  is  correspondingly  diminished.  That 
liquids  represent  a  more  condensed  form  of  matter  than  gases  is 
shown  by  the  change  in  volume  which  results  when  a  liquid  is 
vaporized:  thus,  1  cc.  of  water  at  the  boiling  point  when  vapor- 
ized at  the  same  temperature  occupies  a  volume  of  about  1700 
cc.  A  liquid  contains  less  energy  than  a  gas,  since  energy  is  al- 
ways required  to  transform  it  into  the  gaseous  state.  Since  gases 
can  be  liquefied  by  increasing  the  pressure  and  lowering  the  tem- 
perature, and  since  liquids  can  be  vaporized  by  lowering  the 
pressure  and  increasing  the  temperature,  it  is  apparent  that  there 
is  no  generic  difference  between  the  two  states  of  matter. 

Connection  Between  the  Gaseous  and  Liquid  States.  If  a  gas 
is  compressed  isothermally,  its  state  may  change  in  either  of  two 
ways  depending  upon  the  temperature: — (1)  The  volume  at 
first  diminishes  more  rapidly  than  the  pressure  increases,  then  in 
the  same  ratio  and  lastly  more  slowly.  When  the  pressure  attains 
a  very  high  value  the  volume  is  but  slightly  altered.  This  case 
has  already  been  considered  in  the  preceding  chapter.  (2)  The 
volume  changes  more  rapidly  than  the  pressure  until,  when  a  cer- 
tain pressure  is  reached,  the  gas  ceases  to  be  homogeneous,  and  par- 
tial liquefaction  results.  For  a  constant  temperature,  the  pressure 

53 


54  THEORETICAL  CHEMISTRY 

at  which  liquefaction  .occurs  is  invariable  for  a  given  gas,  while  the 
volume  steadily  diminishes  until  liquefaction  is  complete.  Only 
when  the  whole  mass  of  gas  has  been  liquefied  is  it  possible  to 
increase  the  pressure  and  then,  owing  to  the  small  compressibility 
of  liquids,  a  large  increase  in  pressure  is  required  to  produce  a 
slight  diminution  in  volume.  If  the  temperature  is  above  a  cer- 
tain point,  dependent  upon  the  nature  of  the  gas,  the  phenomena 
of  compression  will  follow  (1);  if  below  this  point,  the  process 
will  follow  (2).  That  a  gas  may  behave  in  either  of  the  above 
ways  was  first  clearly  recognized  by  Andrews*  in  1869,  in  connec- 
tion with  his  experiments  on  the  liquefaction  of  carbon  dioxide. 
He  found  that  if  carbon  dioxide  was  compressed,  keeping  the 
temperature  at  0°,  the  volume  changes  more  rapidly  than  the 
pressure,  liquefaction  resulting  when  a  pressure  of  35.4  atmos- 
pheres was  reached.  As  the  temperature  was  raised,  he  found 
that  a  higher  pressure  was  required  to  liquefy  the  gas,  until  at 
temperatures  above  30°. 92  it  was  no  longer  possible  to  condense 
the  gas  to  the  liquid  state.  The  temperature  above  which  it  was 
no  longer  possible  to  liquefy  the  gas  he  termed  the  critical  tem- 
perature. In  like  manner  the  pressure  required  to  liquefy  the 
gas  at  the  critical  temperature,  he  termed  the  critical  pressure, 
and  the  volume  occupied  by  the  gas  or  the  liquid  under  these 
conditions  he  called  the  critical  volume. 

Isothermals  of  Carbon  Dioxide.  The  results  of  Andrew's 
experimentsf  on  the  liquefaction  of  carbon  dioxide  are  shown  in 
Fig.  11,  in  which  the  ordinates  represent  pressures,  and  the  abscis- 
sae the  corresponding  volumes,  at  constant  temperature.  The 
curves  obtained  by  plotting  volumes  against  pressures  at  constant 
temperatures  are  called  isothermals.  For  a  gas  which  follows 
Boyle's  law,  the  isothermals  will  be  a  series  of  equilateral  hy- 
perbolas. This  condition  is  approximately  fulfilled  by  air,  for 
which  three  isothermals  are  given  in  the  diagram.  At  48°  the 
isothermal  for  carbon  dioxide  is  nearly  hyperbolic,  but  as  the 
temperature  becomes  lower,  the  isothermals  deviate  more  and 
more  from  those  for  an  ideal  gas.  At  the  critical  temperature, 
30°.92,  the  curve  is  almost  horizontal  for  a  short  distance,  showing 
that  for  a  very  slight  change  in  pressure  there  is  an  enormous 
shrinkage  in  volume.  At  still  lower  temperatures,  21°.  1  and  13°.  1, 

*  Trans.  Roy.  Soc.,  159,  583  (1869). 
t  loc.  cit. 


LIQUIDS 


55 


the  horizontal  portions  of  the  curves  are  much  more  pronounced, 
indicating  that  during  liquefaction  there  is  no  change  in  pressure. 
When  liquefaction  is  complete,  the  curves  rise  abruptly,  showing 


Carbon  Dioxide  — 


Air 


Volume 
Fig.   11 

that  the  change  in  volume  is  extremely  small  for  a  large  increase 
in  pressure;  in  other  words  the  liquefied  gas  possesses  a  small 
coefficient  of  compressibility.  At  any  point  within  the  para- 
bolic area,  indicated  by  the  dotted  line  ABC,  both  vapor  and 
liquid  are  coexistent;  at  any  point  outside,  only  one  form  of  mat- 


56  THEORETICAL  CHEMISTRY 

ter,  either  liquid  or  vapor,  is  present.  Andrew's  experiments  show 
that  there  is  no  fundamental  difference  between  a  gas  and  a  liquid. 
It  is  apparent  from  the  diagram  that  when  carbon  dioxide  is  sub- 
jected to  great  pressures  above  its  critical  temperature,  it  behaves 
more  like  a  liquid  than  a  gas;  in  fact  it  is  difficult  to  determine 
whether  a  highly  compressed  gas  above  its  critical  temperature 
should  be  classified  as  a  gas  or  as  a  liquid. 

Van  der  Waals'  Equation  and  the  Continuity  of  the  Gaseous 
and  Liquid  States.  In  the  preceding  chapter  we  have  learned 
that  the  fundamental  gas  equation, 

pv  =  RT, 

is  only  strictly  applicable  to  an  ideal  gas,  and  that  the  behavior 
of  actual  gases  is  represented  with  considerable  accuracy,  even  at 
high  pressures,  by  the  equation  of  van  der  Waals, 


If  this  equation  be  arranged  in  descending  powers  of  v,  we  have 


This  being  a  cubic  equation  has  three  possible  solutions,  each 
value  of  p  affording  three  corresponding  values  of  v,  a,  b,  R  and  T 
being  treated  as  constants.  The  three  roots  of  this  equation  are 
either  all  real,  or  one  is  real  and  two  are  imaginary,  depending 
upon  the  values  of  the  constants.  That  is  to  say,  a^onejempera- 
ture  and  pressure  the  values  of  a  a,ndj>  may  be  such^that  v  has 
three  real  values,  wEfle  at  another  temperature  and  pressureTl; 
may  have  one  real  and  two  imaginary  values.  In  the  accompany- 
ing diagram,  Fig.  12,  a  series  of  graphs  of  the  equation  for  different 
values  of  T  is  given.  It  will  be  observed  that  these  curves  bear 
a  striking  resemblance  to  the  isotherms  of  carbon  dioxide  estab- 
lished by  the  experiments  of  Andrews.  In  the  case  of  the  theo- 
retical curves  there  are  no  sudden  breaks  such  as  appear  in  the 
actual  discontinuous  passage  from  the  gaseous  to  the  liquid  state. 
Instead  of  passing  from  B  to  D  along  the  wavelike  path  BaCbD, 
experiment  has  shown  that  the  substance  passes  directly  from  the 
state  B  to  the  state  D  along  the  straight  line  BD.  It  is  here 
that  van  der  Waals'  equation  fails  to  apply.  As  has  been  pointed 


LIQUIDS 


57 


out,  the  substance  between  these  two  points  is  not  homogeneous, 
being  partly  gaseous  and  partly  liquid.  Attempts  have  been 
made  to  realize  the  portion  of  the  curve  BaCbD  experimentally. 
By  studying  supersaturated  vapors  and  superheated  liquids,  it 
has  been  found  possible  to  follow  the  theoretical  curve  for  short 


Volume 
Fig.  12 

distances  between  B  and  D  without  discontinuity,  but  owing  to 
the  instability  of  the  substance  in  this  region,  it  is  evident  that 
the  complete  isothermal  and  continuous  transformation  of  a  gas 
into  a  liquid  can  never  be  effected.  Attention  has  been  called  by 
van  der  Waals  to  the  fact,  that  in  the  surface  layer  of  a  liquid,  where 
unique  conditions  prevail,  it  is  quite  possible  that  such  unstable 
states  may  exist,  and  that  there  the  transition  from  liquid  to  gas 


58  THEORETICAL  CHEMISTRY 

may  in  reality  be  a  continuous  process.  The  diagram  shows  that 
as  T  increases,  the  wave-like  portion  of  the  isothermals  becomes 
less  pronounced  and  eventually  disappears,  when  the  points  B, 
C  and  D  coalesce.  At  this  point  the  three  roots  of  the  equa- 
tion become  equal,  the  volume  of  the  liquid  becoming  identical 
with  the  volume  of  the  gas.  The  substance  at  this  point  is  in  the 
critical  condition.  Since  under  these  conditions  the  three  roots 
of  the  equation, 

/,    .   RT\       .  a         ab      n 
vz  —  [b  H  ---  }v2  +  -v  --  =  0 
\          p  I         P         P 

are  equal,  we  may  write  vi  =  v2  =  vs  =  vc,  the  subscript  c  indicat- 
ing the  critical  state.  Then  equation  (1)  must  be  equivalent  to 

(v  -  vc)s  =  vs  -  3  vcv*  +  3  ve*v  -  v*  =  0.  (2) 

Equating  the  corresponding  coefficients  of  equations  (1)  and  (2), 
we  have 


3vc=b+    —  ,  (3) 

PC 


and 


Ve'  =      ' 


Dividing  equation  (5)  by  equation  (4),  we  have 

vc  =  3  6,  (6) 

and  substituting  this  value  in  equation  (4),  we  obtain 

a  /_, 

PC  =  27^'  (7) 

Lastly,  substituting  the  values  of  vc  and  pc,  given  by  equations 
(6)  and  (7),  in  equation  (3),  we  have 

T.  =  — •  (8) 

Therefore, 

T  =  o  R-  (9) 

1  c  O 


LIQUIDS 


59 


Or  expressing  the  constants  a,  b  and  R  in  terms  of  the  critical 
values  of  pressure,  temperature  and  volume,  we  have 

a  =  3  pcvc2, 


and 


'-I 

P  _  8  pcvc 
~~ 


(10) 
(11) 
(12) 


By  means  of  equations  (6),  (7)  and  (8)  it  is  possible  to  calculate 
the  critical  constants  of  a  gas  when  the  constants  a  and  b  of  van 
der  Waals'  equation  are  known.  Conversely  by  means  of  equa- 
tions (10)  and  (11),  the  values  of  a  and  6  can  be  calculated  when  the 
critical  data  are  given. 

The  calculated  values  of  the  constants,  a  and  6,  for  several  gases 
are  given  in  the  accompanying  table.  In  calculating  these  values 
the  pressure  was  measured  in  atmospheres,  and  the  gram-molec- 
ular volume  in  cubic  centimeters.  Obviously  the  value  of  b  must 
be  multiplied  by  10~3  and  that  of  a  by  10~6,  if  the  volume  is  meas- 
ured in  liters. 


VALUES  OF  VAN  DER  WAALS'  CONSTANTS 


Substance. 

Formula. 

b. 

a. 

Hydrogen 

H2 

23 

0  19  X 

10* 

Oxygen 

O2 

31  6 

1  36 

Nitrogen 

N2 

37  3 

1  31 

Carbon  dioxide 

CO2 

42  8 

3  61 

Carbon  monoxide  

CO 

38.6 

1  43 

Sulphur  dioxide  

SO2 

56.5 

6  69 

Chlorine  

C12 

46.1 

5.35 

Hydrogen  chloride  

HC1 

40.9 

3.81 

Ammonia 

NH3 

36  4 

4  05 

Water 

H2O 

33.2 

5  87 

Ethane 

69.9 

6  0 

Benzene  .  . 

C6H6 

120.3 

18.71 

*  Zeit.  phys.  Chem.  69,  52  (1910). 

It  has  been  pointed  out  that  intermolecular  attraction  depends 
not  only  on  the  size  of  the  molecules  but  also  on  their  constitution. 
Those  molecules  which  are  made  up  of  a  large  number  of  atoms 
exert  a  greater  mutual  attraction  than  molecules  of  the  same  mass 
but  composed  of  a  smaller  number  of  atoms.  This  is  shown  by 


60 


THEORETICAL  CHEMISTRY 


the  data  of  the  foregoing  table  in  which  the  value  of  a  for  ethane, 
6.0  X  106,  is  over  four  times  greater  than  the  corresponding  value 
of  a  for  oxygen,  1.36  X  106,  although  their  molecular  weights  are 
nearly  the  same. 

Molecular  Diameters.  It  has  been  claimed  by  Richards* 
that  the  volume  corresponding  to  the  constant  b  is  approximately 
1.2  times  the  volume  of  one  gram-molecule  of  the  liquefied  gas 
at  its  boiling-point,  and  that  it  may  be  regarded  as  representing 
the  actual  volume  of  the  molecules  themselves.  If  this  be  true, 
we  may  readily  compute  the  so-called  ''molecular  diameters." 
To  do  this,  it  is  only  necessary  to  divide  the  value  of  b,  expressed 
in  cubic  centimeters  per  gram-molecular  weight,  by  Avogadro's 
constant,  and  then  extract  the  cube  root  of  the  quotient.  The 
values  of  the  molecular  diameters  calculated  in  this  manner  for 
several  gases  are  given  in  the  following  table,  together  with  the 
corresponding  values  computed  by  means  of  an  equation  derived 
from  measurements  of  the  mean  free  path  of  a  molecule,  i.e., 
from  the  distance  a  molecule  travels  before  colliding  with  another 
molecule.  The  agreement  between  the  results  obtained  by  the 
two  methods  is  most  satisfactory. 

MOLECULAR  DIAMETERS 


Substance. 

Formula. 

.  Molec.  Diam. 
(from  b). 

Molec.  Diam. 
(from  mean  free  path)  . 

Hydrogen  
Oxygen......  
Carbon  dioxide  
Carbon  monoxide  

H2 
02     , 
CO2 
CO 

2.34  X  10-8cm. 
2.92 
3.23 
3.12 

2.4  X  10-*  cm. 
2.97 
3.36 
3.19 

Nitrogen 

N2 

3  15 

3.15 

Ammonia 

NH3 

3.08 

2.97 

Corresponding  States.  If  in  the  equation  of  van  der  Waals, 
the  values  of  p,  v  and  T  be  expressed  as  fractions  of  the  corre- 
sponding critical  values,  we  may  write 


and 


p  =  apc, 
v  =  &vc 

T  =  yTc. 
*  Jour.  Am.  Chem.  Soc.,  36,  629  (1914). 


LIQUIDS  61 

Substituting  these  values  in  the  equation 

(p  +  £)(»-&)  =  RT, 

we  have 


/ 


and  replacing  pc,  vc,  and  Tc  by  their  values  given  in  equations  (6), 
(7)  and  (8),  we  obtain 

(W|)(3/3-l)=8T,  (13) 

which  is  known  as  van  der  Waals'  reduced  equation  of  state. 

In  this  equation  everything  connected  with  the  individual 
nature  of  the  substance  has  vanished,  thus  making  it  applicable 
to  all  substances  in  the  liquid  or  gaseous  state  in  the  same  way 
that  the  fundamental  gas  equation  is  applicable  to  all  gases,  irre- 
spective of  their  specific  nature.  It  has  been  shown,  however, 
that  the  equation  is  not  entirely  trustworthy  and  at  best  can  be 
considered  as  little  more  than  a  rough  approximation.  The 
chief  point  to  be  observed  in  connection  with  this  equation  is 
that  whereas  for  gases,  the  corresponding  values  of  temperature, 
pressure  and  volume,  measured  in  the  ordinary  units,  may  be 
compared,  it  is  necessary  in  the  case  of  liquids  to  make  the  com- 
parison under  corresponding  states.  According  to  van  der  Waals, 
two  substances  are  in  corresponding  states  when  their  pressures 
are  proportional  to  their  critical  pressures,  their  volumes  to  their 
critical  volumes  and  their  temperatures  to  their  critical  tem- 
peratures; or,  in  other  words,  two  substances  are  in  cor- 
responding states  when  their  reduced  pressures,  volumes  and 
temperatures  are  equal.  The  reduced  equation  of  state  has  been 
investigated  by  Young*  for  about  thirty  substances  at  the  same 
reduced  pressure,  0.08846.  The  following  table  is  compiled  from 
the  data  given  by  Young. 

*  Phil.  Mag.  V  33,  153  (1892). 


62 


THEORETICAL  CHEMISTRY 


REDUCED  ELITES  IN  CORRESPONDING  STATES 

Ratio  of  Pressure  to  Critical  Pressure  =  0.08846 


Substance. 

T/TC 

V  (liquid)  /  Vc 

F(gas)/Fc 

IMethyl  alcohol                           .    .  . 

0.7734 

0  3973 

34  35 

Ethyl  alcohol                        

0.7794 

0  4061 

32  15 

Propyl  alcohol                  

0.7736 

0  4002 

30  85 

Methyl  acetate       

0.7445 

0  3989 

30  15 

Ethyl  acetate  

0.7504 

0.4001 

30  25 

Propyl  acetate  

0.7541 

0.3985 

30  35 

Acetic  acid 

0.7624 

0  4100 

25  4 

Ether 

0.7380 

0  4030 

28  3 

Benzene                                            .    . 

0.7282 

0  4065 

28  3 

Chlorbenzene                      .       

0.7345 

0  4028 

28  5 

Brombenzene                     

•  0.7343 

0  4024 

28  3 

lodobenzene  

0.7337 

0.4020 

28  3 

The  data  of  the  foregoing  table  affords  ample  confirmation  of 
the  validity  of  van  der  Waals  rule  that:  "  When  the  absolute 
temperatures  of  two  substances  are  proportional  to  their  absolute 
critical  temperatures,  their  vapor  pressures  will  be  proportional 
to  their  critical  pressures,  and  their  orthobaric*  volumes,  both  as 
liquid  and  vapor,  will  be  proportional  to  their  critical  volumes." 

Berthelot's  Equation.  Numerous  modifications  of  the  equation 
of  van  dei*  Waals  have  been  proposed  from  time  to  time,  but  of 
these  only  one  need  be  considered  here.  As  has  already  been 
pointed  out,  a  well-defined  relation  is  known  to  exist  between  the 
constants  a  and  b  of  any  particular  gas  and  its  critical  constants, 
An  equation  derived  by  D.  Berthelot,  f  in  which  a  and  b  are  ex- 
pressed in  terms  of  the  critical  constants,  has  been  found  to  repre- 
sent the  behavior  of  pure  gases  with  a  high  degree  of  accuracy 
over  an  extended  range  of  pressures.  Berthelot's  equation  may 
be  written  in  the  following  form  :  — 


In  this  equation  pc  and  Tc  are  the  critical  pressure  and  the  critical 
temperature  respectively.  Inspection  of  the  equation  shows 
that  as  T  increases  or  as  p  decreases,  it  approaches  the  simple 

*  The  volume  of  a  liquid  at  a  given  temperature  and  under  a  pressure 
equal  to  the  vapor  pressure  is  called  the  orthobaric  volume. 
t  Jour.  Phys.  Ill  8,  263  (1899). 


LIQUIDS  63 

equation,  pv  =  RT,  expressing  the  behavior  of  an  ideal  gas. 
The  values  of  the  ratio,  pv/RT,  calculated  by  means  of  equation 
(14)  agree  closely  with  those  determined  experimentally,  pro- 
vided the  gases  are  neither  associated  nor  dissociated,  and  that 
they  are  sufficiently  removed  from  their  critical  points. 

The  most  important  application  of  Berthelot's  equation  is  in 
the  precise  determination  of  atomic  weights  by  the  physical 
method  (see  p.  17).  The  approximate  value  of  the  molecular 
weight  M,  of  m  grams  of  any  gas  may  be  calculated  by  the  equa- 
tion 

r>m 

M  =  m—>  (15) 

pv 

or,  since  d  =  m/v,  where  d  is  the  density  of  the  gas,  equation  (15) 
may  be  written  in  the  form 

M  =  d—-  (16) 

P 

If  we  wish  to  determine  the  exact  molecular  weight  of  a  gas,  how- 
ever, it  is  necessary  to  introduce  a  correction  for  the  deviation 
from  the  gas  laws.  To  do  this,  we  make  use  of  the  equation  of 
Berthelot  instead  of  the  simple  equation,  pv  =  R  T,  and  obtain 
the  equation 


or,  letting  A  denote  the  expression  within  brackets,  we  have 

M  =  d—(l  +  Ap).  (18) 

From  this  value  of  M,  the  exact  atomic  weight  can  be  calculated 
by  dividing  by  the  number  of  atoms  contained  in  the  molecule. 
This  method  of  calculating  exact  atomic  weights  is  known  as 
the  "  method  of  critical  constants." 

Owing  to  the  fact  that  the  deviation  from  the  simple  gas  laws 
becomes  less  and  less  as  the  pressure  is  lowered,  it  is  evident  that 
the  exact  value  of  the  molecular  weight  of  a  gas  can  also  be  cal- 
culated directly  from  its  density,  provided  this  can  be  determined 
in  the  neighborhood  of  zero-pressure.  The  determination  of  this 
so-called  "  limiting  density  "  can  be  made  with  a  high  degree  of 
accuracy,  if  the  values  of  the  density  of  the  gas  at  two  or  more 


64  THEORETICAL  CHEMISTRY 

pressures  are  known.  To  determine  the  value  of  the  limiting 
density,  the  values  of  the  ratio  of  density  to  pressure  are  plotted 
as  ordinates  against  the  corresponding  values  of  the  pressure  as 
abscissas,  and  where  the  extrapolated  curve  cuts  the  axis  of  zero- 
pressure,  the  value  of  d/p,  gives  the  value  of  the  limiting  density. 
The  method  of  limiting  densities  has  been  found,  in  general,  to  give 
more  trustworthy  results  than  the  method  of  critical  constants, 
owing  to  the  fact  that  the  density  of  a  gas  can  be  measured  with 
greater  accuracy  than  its  critical  constants. 

Liquefaction  of  Gases.  The  history  of  the  liquefaction  of 
gases  has  for  a  long  time  been  regarded  as  one  of  the  most  inter- 
esting chapters  of  physical  science.  Among  the  first  success- 
ful workers  in  this  field  was  Faraday.*  He  liquefied  practically 
all  of  the  gases  which  condense  under  moderate  pressures  and  at 
not  very  low  temperatures.  The  apparatus  used  by  Faraday 
consisted  of  an  inverted  V-shaped  tube,  in  one  end  of  which  was 
placed  some  solid  which  would  liberate  the  desired  gas  on  heating, 
while  the  other  end  was  sealed  and  immersed  in  a  freezing  mixture. 
When  the  substance  had  been  heated  long  enough  to  liberate 
considerable  gas,  the  pressure  became  sufficiently  high  to  cause 
the  gas  to  liquefy  at  the  temperature  of  the  other  end  of  the  tube. 
Thus,  when  chlorine  hydrate  was  heated  in  the  tube,  the  liberated 
chlorine  condensed  as  a  yellow  liquid. 

In  1834,  Thilorierj  succeeded  in  liquefying  carbon  dioxide  in 
quite  large  amounts  by  the  use  of  a  new  form  of  apparatus.  In 
connection  with  his  experiments  on  liquid  carbon  dioxide,  he 
observed  that  when  it  was  allowed  to  vaporize,  enough  heat 
was  absorbed  to  lower  the  temperature  below  its  freezing-point, 
solid  carbon  dioxide  being  obtained.  He  discovered  that  a  mix- 
ture of  solid  carbon  dioxide  and  ether  was  a  powerful  refrig- 
erant, and  that  under  diminished  pressure  the  mixture  gave 
temperatures  ranging  from  — 100°  C.  to  —  110°C.  This  mix- 
ture is  known  today  as  Thilorier's  mixture. 

Faraday  J  undertook  the  liquefaction  of  the  so-called  permanent 
gases  in  1845.  In  this  second  series  of  experiments,  he  employed 
higher  pressures  than  in  his  earlier  experiments,  and  also  made 
use  of  the  newly  discovered  Thilorier  mixture  as  a  refrigerant. 

*  Phil.  Trans.,  113,  189  (1823). 
t  Lieb.  Ann.,  30,  122  (1839). 
J  Phil.  Trans.,  135,  155  (1845). 


>  LIQUIDS  65 

He  was  partially  successful  in  his  attempt  to  liquefy  the  hitherto 
noneondensible  gases.  He  liquefied  ethylene,  phosphine  and 
hydrobromic  acid  and  also  solidified  ammonia,  cyanogen,  and 
nitrous  oxide.  He  failed  to  liquefy  hydrogen,  oxygen,  nitrogen, 
nitric  oxide  and  carbon  monoxide.  No  further  advance  in  the 
liquefaction  of  gases  was  made  until  the  year  1869,  when  Andrews 
pointed  out  the  importance  of  cooling  the  gas  below  its  critical 
temperature.  This  discovery  explained  why  so  many  of  the 
earlier  experiments  had  failed,  and  opened  the  way  to  the  brilliant 
successes  of  the  latter  part  of  the  nineteenth  century. 

In  1877,  Cailletet*  and  Pictet,f  working  independently,  suc- 
ceeded in  liquefying  oxygen.  Cailletet  subjected  the  gas  to  a 
pressure  of  about  300  atmospheres  using  boiling  sulphur  dioxide 
as  a  refrigerant.  The  gas  was  further  cooled  by  suddenly  releas- 
ing the  pressure  and  allowing  it  to  expand.  In  addition  to 
oxygen  he  also  liquified  air,  nitrogen  and  possibly  hydrogen. 

Shortly  afterward  in  1883,  the  Polish  scientists,  Wroblewski 
and  Olszewski,  J  published  an  account  of  their  interesting  and 
highly  important  work.  In  their  experiments  they  subjected 
the  gas  to  be  liquefied  to  high  pressure,  and  simultaneously 
cooled  it  to  a  very  low  temperature.  Among  the  refrigerants 
used  by  them  was  liquid  ethylene,  which  was  allowed  to  boil 
off  under  diminished  pressure,  giving  a  temperature  of  —  130°  C. 
At  this  temperature,  a  pressure  of  only  20  atmospheres  was 
sufficient  to  condense  oxygen  to  the  liquid  state.  Having 
liquefied  oxygen,  nitrogen,  air  and  carbon  monoxide,  and  hav- 
ing determined  the  boiling-points  of  these  gases  under  atmos- 
pheric pressure,  they  proceeded  to  use  these  liquefied  gases 
as  refrigerants,  allowing  them  to  boil  off  under  diminished 
pressure,  thus  obtaining  temperatures  as  low  as  —200°  C.  A 
very  small  amount  of  liquid  hydrogen  was  obtained  in  this  way. 
Subsequent  attempts  by  these  same  experimenters  to  liquefy 
hydrogen,  while  not  much  more  successful  than  their  former 
attempts,  enabled  them  to  determine  its  boiling-point.  Shortly 
after  the  publication  of  the  first  papers  of  Wroblewski  and 
Olszewski,  Dewar§  devised  a  new  form  of  apparatus  for  lique- 

*  Compt.  rend.,  85,  1217  (1877). 
t  Ibid.,  85,  1214,  1220  (1877). 
J  Wied.  Ann.,  20,  243  (1883). 
§  Proc.  Roy.  Inst.,  1886,  550. 


66  THEORETICAL   CHEMISTRY 

fying  air,  oxygen  and  nitrogen  on  a  comparatively  large  scale. 
He  also  introduced  the  well-known  vacuum-jacketed  flasks  and 
tubes  which  greatly  facilitated  carrying  out  experiments  with 
liquefied  gases. 

In  1895,  Linde  in  Germany  and  Hampson  in  England  simul- 
taneously and  independently  constructed  machines  for  the  lique- 
faction of  air  in  large  quantities.  In  the  method  devised  by 
these  experimenters  the  air  is  not  subjected  to  a  preliminary 
cooling,  produced  by  the  rapid  evaporation  of  a  liquefied  gas 
under  diminished  pressure,  as  in  the  methods  of  Wroblewski 
and  Olszewski.  In  the  Linde  liquefier,  the  air  is  compressed 
to  about  200  atmospheres.  It  is  then  passed  through  a  cham- 
ber containing  anhydrous  calcium  chloride  to  remove  the  greater 
part  of  the  moisture,  after  which  it  is  cooled  by  allowing  it 
to  circulate  through  a  coiled  pipe  immersed  in  a  freezing  mixture. 
Nearly  all  of  the  moisture  remaining  in  the  air  is  deposited  on  the 
walls  of  the  pipe  in  the  form  of  frost.  The  air  then  enters  a  long 
spiral  tube  jacketed  with  a  non-conducting  material,  and  is  there 
allowed  to  expand  to  a  pressure  of  about  15  atmospheres. 
During  this  expansion  the  temperature  of  the  air  is  appreciably 
lowered.  When  the  air  has  traversed  the  spiral  tube,  it  is  still 
further  cooled  by  allowing  it  to  expand  to  a  pressure  equal  to  that 
of  the  atmosphere.  The  air  which  has  been  thus  cooled  is  then 
passed  backward  through  the  annular  space  between  the  spiral 
tube  and  a  concentric  jacket,  thus  cooling  the  entering  portion  of 
air.  Consequently  this  next  portion  of  air  expands  from  a  lower 
initial  temperature,  and  the  cooling  effect  is  increased.  In  like 
manner,  when  this  cooler  air  passes  backward  it  cools  still  further 
the  next  succeeding  portion,  and  eventually  the  temperature  is 
reduced  sufficiently  to  cause  a  small  amount  of  the  air  to  liquefy  as 
it  issues  from  the  end  of  the  spiral  tube.  The  remaining  portion  of 
the  air  which  has  not  been  liquefied,  passes  backward  through  the 
annular  tube  and  cools  the  following  portion  to  a  still  greater 
extent,  causing  a  larger  proportion  to  liquefy  on  expansion.  With 
a  3-horse-power  machine,  a  continuous  supply  of  0.9  liter  per  hour 
can  be  obtained.  Further  improvements  in  this  apparatus  have 
been  made  by  Dewar  and  Hampson,  and  by  means  of  it  Dewar's 
brilliant  successes  in  the  liquefaction  of  gases  have  been  achieved. 
The  most  efficient  apparatus  for  the  liquefaction  of  air  and  other 


LIQUIDS 


67 


gases  is  that  developed  by  Claude.*  The  essential  features  of 
his  liquefier  are  shown  in  Fig.  13.  The  air  is  first  compressed  to 
40  atmospheres  pressure  by  means  of  an  ordinary  compression 
pump  not  shown  in  the  diagram,  the  moisture  and  carbon  dioxide 
being  removed  as  in  Linde's  method.  It  then  enters  the  tube  A, 
which  in  reality  is  of  a  spiral  form,  and  divides  at  B.  A  portion 
enters  the  cylinder  D  through  a  valve  chest  similar  to  that  in  a 
steam  engine,  forcing  out  the  piston  and  causing  the  wheel,  W,  to 


Liquid  Air 
40  atmoe.,-140 


Fig.   13 

revolve,  thereby  doing  work  and  simultaneously  cooling  the  air. 
The  cooled  air  escapes  from  the  valve  chest  and  circulates  through 
the  liquefying  chamber  L,  where  it  causes  the  portion  of  compressed 
air  entering  at  B  to  liquefy.  It  then  issues  from  the  liquefier  and 
traverses  M,  cooling  the  entering  portion  of  air  in  A,  and  finally 
returns  to  the  compressor.  The  pressure  of  the  air  when  it  issues 
from  D  is  almost  atmospheric,  but  its  temperature  is  below 
— 140°  C.  About  twenty-five  per  cent  of  the  power  consumed 
in  compression  is  regained  by  the  motor.  The  apparatus  pro- 
duces about  1  liter  of  liquid  air  per  horse-power  hour.  By  means 
of  this  improved  apparatus,  based  upon  the  regenerative  principle, 
all  known  gases  have  now  been  liquefied,  the  last  to  succumb  being 

*  Compt.  rend.,  II,  500  (1900);  I,  1568  (1902);  II,  762,  823  (1905). 


68 


THEORETICAL  CHEMISTRY 


helium  which  was  liquefied,  in  1908,  by  Kammerlingh  Onnes  in  the 
Leyden  cryogenic  laboratory.  The  subjoined  table  gives  the 
critical  data,  together  with  the  boiling  and  freezing  temperatures, 
of  some  of  the  more  common  gases. 

PHYSICAL  CONSTANTS  OF  SOME  OF  THE  PRINCIPAL  GASES  * 


Gas 

Density 
(0°,760mm.) 

Grit. 
Density. 

Melting 
Point. 

Boiling  Point 
(760  mm.) 

Grit. 
Temp. 

Grit.  Press. 
(Atmos.). 

He.  . 

0  .  1782 

0.07 

4.5°Abs. 

5.0°Abs. 

2.3 

H2  

0.08987 

0.033 

14.1°  Abs. 

20.4 

31 

13.4 

Ne.  . 

0.9002 

30 

53 

55 

29 

N2 

1.2507 

0.311 

62.5 

77.3 

127 

33.0 

CO 

1.2504 

66 

83 

133.5 

35.5 

O2 

1.4290 

0  .  162 

46 

90.1 

154.2 

50.8 

A  

1.7809 

0.509 

85.1 

87.2 

155.6 

52.9 

NO  

1.3402 

112.5 

122.5 

179.5 

71.2 

CH4  

0.7168 

89 

108.3 

191.2 

54.9 

Kr  

3.708 

104 

121.3 

210.5 

54.3 

Xe  

5.851 

1.15" 

133 

163 

287.7 

57.2 

C02  

-1.9768 

0.448 

216.3 

194.7 

304.1 

73.0 

HC1  

1.6394 

0.41 

161.6 

189.9 

324.8 

83.6 

NH3  

0.7708 

194.8 

239.5 

405.3 

109.6 

Cl,  

3.1674 

171.0 

239.4 

419 

93.5 

S02  

2.9266 

0.513' 

200.7 

263 

430.0 

78.0 

H2S  

1.5392 

190.0 

212.8 

373.4 

89.3 

PH3  

1.5293 

140.5 

186.6 

324.3 

64.5 

*  Baume,  Jour.  Chim.  Phys.  9,  282  (1911). 

Vapor  Pressure  of  Liquids.  According  to  the  kinetic  theory 
there  is  a  continuous  flight  of  particles  of  vapor  from  the  surface 
of  a  liquid  into  the  free  space  above  it.  At  the  same  time  the 
reverse  process  of  condensation  of  vapor  particles  at  the  surface 
of  the  liquid  is  taking  place.  Eventually  a  condition  of  equilib- 
rium will  be  established  between  the  liquid  and  its  vapor,  when 
the  rate  of  escape  will  be  exactly  counterbalanced  by  the  rate  of 
condensation  of  vapor  particles:  the  vapor  is  then  said  to  be  sat- 
urated. The  pressure  exerted  by  the  vapor  of  a  liquid  when  equi- 
librium has  been  attained  is  known  as  its  vapor  pressure.  The 
equilibrium  between  a  liquid  and  its  vapor  is  dependent  upon  the 
temperature.  For  every  temperature  below  the  critical  temper- 
ature, there  is  a  certain  pressure  at  which  vapor  and  liquid  may 
exisMn  equilibrium  in  all  proportions;  and  conversely  for  every 
pressure  below  the  critical  pressure,  there  is  a  certain  temperature 
at  which  vapor  and  liquid  may  exist  in  equilibrium  in  all  propor- 


LIQUIDS  69 

tions.  This  latter  temperature  is  termed  the  boiling-point  of  the 
liquid.  The  vapor  pressure  of  a  liquid  may  be  measured  directly 
by  placing  a  portion  of  it  above  the  mercury  in  the  vacuum  of  a 
barometer  tube,  heating  to  the  desired  temperature,  and  observ- 
ing the  depression  of  the  mercury  column.  This  is  known  as  the 
static  method.  It  is  open  to  the  objection  that  the  presence  of 
volatile  impurities  in  the  liquid  causes  too  great  depression  of  the 
mercury  column,  the  vapor  pressure  of  the  impurity  adding  itself 
to  that  of  the  liquid  whose  vapor  pressure  is  sought.  A  better 
method  for  the  measurement  of  vapor  pressure  is  that  known  as 
the  dynamic  method.  In  this  method  the  pressure  is  maintained 
constant  and  the  boiling  temperature  is  determined  with  an  ac- 
curate thermometer.  The  boiling  temperatures  corresponding 
to  various  pressures  may  be  measured,  provided  we  have  a  suit- 
able device  for  changing  and  measuring  the  pressure.  The  results 
obtained  by  the  static  and  dynamic  methods  agree  closely  if  the 
liquid  is  pure,  but  if  volatile  impurities  are  present  the  results 
obtained  by  the  dynamic  method  are  more  trustworthy.  A 
method  for  the  measurement  of  vapor  pressure  due  to  James 
Walker*  is  of  considerable  interest.  In  this  method,  a  currerit 
of  pure  dry  air  is  bubbled  through  a  weighed  amount  of  the  liquid 
whose  vapor  pressure  is  to  be  determined.  The  liquid  is  main- 
tained at  constant  temperature  and  its  loss  in  weight  is  observed. 
In  passing  through  the  liquid  the  air  will  absorb  an  amount  of 
vapor  directly  proportional  to  the  vapor  pressure  of  the  liquid. 
If  1  mol  of  liquid  is  absorbed  by  v  liters  of  air,  then  we  have 

pv  =  RT, 

where  p  is  the  vapor  pressure  of  the  liquid,  and  T  its  temperature. 
If  v\  is  the  volume  of  air  which  absorbs  g  grams  of  the  vapor  of 
the  liquid  whose  molecular  weight  is  M,  then 


or 

P-ifc**1-  (19) 

In  this  equation,  Vi  denotes  the  total  volume  containing  g  grams 
of  the  liquid  in  the  form  of  vapor,  or  in  other  words  it  represents 

*  Zeit.  phys.  Chem.,  2,  602  (1888). 


70 


THEORETICAL  CHEMISTRY 


the  air  and  vapor  together.  Since  the  volume  of  the  air  is  in 
general  so  much  greater  than  that  of  the  vapor,  vi  may  be  taken 
as  that  of  th'e  air  alone. 

Other  methods  for  measuring  vapor  pressure  will  be  considered 
in  a  later  chapter. 

Law  of  Cailletet  and  Mathias.  If  the  densities  of  a  liquid  and 
its  saturated  vapor  are  plotted  against  temperature,  a  curve  of  the 
form  shown  in  Fig.  14  is  obtained.  This  curve  shows  that  as  the 


Critical  Point 


30 


§20 


§10 


0.2 


0.4  0.6 

Fig.  14 


0.8 


temperature  is  raised  the  densities  of  the  liquid  and  vapor  phases 
approach  each  other  and,  at  the  critical  temperature,  become 
equal.  Cailletet  and  Mathias*  pointed  out  that  the  mean  of  the 
densities  of  any  substance  in  the  state  of  liquid  and  saturated 
vapor,  when  plotted  against  the  corresponding  temperatures,,  lie 
on  a  straight  line.  If  di  and  dv  denote  the  densities  of  liquid 
and  vapor  respectively,  the  relation  of  Cailletet  and  Mathias' 
may  be  expressed  mathematically  by  the  equation, 


dv 


bT, 


(20) 


where  a  and  b  are  constants.  This  relation  is  sometimes  called 
the  law  of  the  rectilinear  diameter.  It  holds  fairly  well  for  a 
number  of  substances,  but  with  others  it  is  found  that  the  mean 
of  the  densities  of  the  two  phases  is  not  strictly  a  linear  function 
of  the  temperature.  In  general  it  may  be  said  that  the  relation 

*  Compt.  rend.  102,  1202  (1886);   104,  1563  (1887), 


LIQUIDS 


71 


holds  in  the  immediate  vicinity  of  the  critical  temperature  better 
than  at  lower  temperatures.  The  law  of  Cailletet  and  Mathias 
has  proven  useful  in  determining  the  value  of  the  critical  density, 
a  quantity  which  frequently  cannot  be  measured  directly. 

Boiling-Point  and  Critical  Temperature.  An  interesting  rela- 
tion has  been  pointed  out  by  Guldberg*  and  Guye.  |  These 
two  investigators  have  shown  that  the  absolute  boiling  tempera- 
ture of  a  liquid  is  about  two-thirds  of  its  critical  temperature. 
That  this  empirical  relation  holds  for  a  variety  of  different  sub- 
stances is  shown  in  the  accompanying  table. 

RELATION  OF  BOILING-POINT  TO  CRITICAL  TEMPERATURE 


Substance. 

Tb 

Tc 

Tb/Te 

Oxygen 

90° 

155° 

0  58 

Chlorine 

240° 

414° 

0  58 

Sulphur  dioxide         .        

263° 

429° 

0  61 

Ethyl  ether     

308° 

467° 

0  66 

Ethyl  alcohol  

351° 

516° 

0  68 

Benzene 

353° 

562° 

0  63 

Water  

373° 

637° 

0.59 

Phenol 

454° 

691° 

0  66 

The  Law  of  Ramsay  and  Young.  It  was  pointed  out  by  Ram- 
say and  YoungJ  that  if  the  absolute  boiling-points  of  two  closely 
related  liquids,  A  and  B,  are  compared  under  equal  pressures,  the 
following  relation  will  be  found  to  hold  good :  — 


-  TA), 


(21) 


nri  rr\t        i     v  \ 

JL  B  IB 

•where  TA  and  TB  are  the  absolute  boiling-points  of  A  and  B 
under  the  same  pressure,  T'A  and  T'B  the  corresponding  boiling- 
points  under  another  pressure,  and  c  is  a  constant.  This  relation 
has  been  shown  to  be  valid  from  very  low  pressures  up  to  the  crit- 
ical pressure.  The  constant  c  is  practically  zero  for  the  major- 
ity of  closely  related  liquids.  The  law  of  Ramsay  and  Young 
has  been  found  very  useful  in  estimating  the  values  of  the  boiling- 
points  of  different  liquids  over  a  range  of  pressures,  the  correspond- 
ing variation  in  boiling-point  of  a  closely  related  liquid  being  taken 

*  Zeit.  phys.  chem.,  5,  376  (1890). 

f  Bull,  Soc.  Chim.,  (3),  4,  262  (1890). 

J  Phil.  Mag.  V  20,  515  (1885);  21,  33  (1886);  22,  37  (1886). 


72  THEORETICAL  CHEMISTRY 

as  a  standard  of  comparison.  Furthermore,  variation  in  the 
value  of  the  constant  c  may  be  used  as  a  criterion  of  abnormality 
in  the  molecular  state  of  liquids.  A  law  similar  to  that  of  Ram- 
say and  Young  has  recently  been  shown  by  Creighton*  to  apply 
to  fluidity,  surface  tension  and  reaction  pressure. 

Heat  of  Vaporization.  In  order  to  transform  a  liquid  into  a 
vapor,  a  large  amount  of  heat  is  required.  Thus,  when  a  liquid 
is  heated  to  the  boiling-point,  the  volume  must  be  increased 
against  the  pressure  of  the  atmosphere,  external  work  being  done, 
and  when  the  boiling  temperature  is  reached,  the  liquid  must  be 
vaporized.  The  heat  expended  in  causing  the  change  of  physical 
state  is  much  greater  than  that  required  to  expand  the  liquid. 
An  interesting  relation  between  the  heat  of  vaporization  and  the 
absolute  boiling-point  of  a  liquid  was  discovered  by  Trouton.f 
If  T  denotes  the  absolute  boiling-point  and  Lv,  the  heat  of  vapor- 
ization of  1  gram  of  liquid  whose  molecular  weight  is  M,  then  ac- 
cording to  Trouton 

-  21 
-21, 


or  in  words,  the  ratio  of  the  molecular  heat  of  vaporization  to  the 
absolute  boiling  temperature  of  a  liquid  is  constant,  the  numerical 
value  of  the  ratio  being  approximately  21.  This  is  known  as 
Trouton's  law.  While  this  relation  holds  quite  well  for  many 
liquids,  Nernst  has  pointed  out  that  the  constant  varies  with  the 
temperature,  and  has  proposed  two  other  forms  of  the  equation. 
Bingham  has  simplified  the  equations  of  Nernst  to  the  following 
form  :  — 

^  =17  +  0.011  T.  (23) 

While  this  modification  of  the  Trouton  equation  has  been  found 
to  hold  for  a  large  number  of  substances,  there  are  other  substances 
for  which  the  left  side  of  the  equation  has  a  value  greater  than 
that  of  the  right  side.  Bingham  infers  that  where  this  occurs,  the 
substance  in  the  liquid  state  has  a  greater  molecular  weight 
than  it  has  in  the  gaseous  state,  or  in  other  words,  the  liquid  is 
associated.  It  is  evident  that  an  associated  liquid  will  require 
an  expenditure  of  energy  over  and  above  that  required  for  vapori- 

*  Jour.  Franklin  Inst.  193,  647  (1922). 
t  Phil.  Mag.  (5),  18,  54  (1884). 


LIQUIDS 


73 


zation,  to  break  down  the  molecular  complex.  The  difference 
between  the  values  of  the  two  sides  of  the  equation  may  be 
taken  as  a  rough  measure  of  the  degree  of  association. 

The  Internal  Pressure  of  Liquids.  The  importance  of  allowing 
for  the  existence  of  attractive  forces  acting  between  the  molecules 
of  gases  becomes  more  and  more  evident  as  the  pressure  is 
increased.  As  has  been  pointed  out,  the  term  a/v2,  was  intro- 
duced by  van  der  Waals  into  the  equation  expressing  the  simple 
gas  laws  to  correct  for  this  intermolecular  attraction.  If  such 
attractive  forces  did  not  exist,  it  is  obvious  that  much  larger 
pressures  would  be  required  to  bring  about  the  liquefaction  of 
gases  than  is  actually  the  case.  For  example,  it  may  be  shown 
that  if  no  attractive  forces  were  operative  between  the  molecules 
of  water  vapor,  the  pressure  required  to  condense  one  gram-mole- 
cule of  vapor  into  the  volume  occupied  by  one  gram-molecule 
of  liquid  water,  at  room  temperature,  would  be  over  1300  atmos- 
pheres. The  pressure  required  to  condense  one  gram-molecule 
of  a  substance  in  the  gaseous  state  to  the  volume  occupied  by 
one  gram-molecule  in  the  liquid  state,  in  the  absence  of  inter- 
molecular  attraction,  is  called  the  internal  pressure  of  the  liquid. 
The  ultimate  cause  of  intermolecular  attraction  is  still  a  matter 
of  conjecture  and,  although  numerous  equations  have  been  de- 
rived for  the  calculation  of  -the  internal  pressure  of  liquids,  the 
results  thus  far  obtained  are  not  entirely  satisfactory.  The  fol- 
lowing table  gives  the  values  of  the  internal  pressure  of  a  few 
liquids  as  calculated  by  four  different  methods. 

INTERNAL  PRESSURE  OF  LIQUIDS  * 


Substance 

p<t 

^ 
Pit 

P«| 

Pill 

Ether 

1240  .atmos. 

1600  atmos. 

2590  atmos. 

930  atmos. 

Carbon  tetrachloride.  . 
Benzene 

1600 
1770 

2080 
2360 

3480 
3810 

1190 
1300 

Carbon  disulphide.  .  .  . 

2260 

3040 

•3840 

1920 

*  Hildebrand,  Jour.  Am.  Chem.  Soc.,  41,  1071  (1919). 

f  From  heat  of  vaporization. 

t  From  critical  data. 

§  From  coefficients  of  expansion  and  compressibility 

||  From  coefficient  of  expansion. 


74 


THEORETICAL  CHEMISTRY 


Surface  Tension.  The  attraction  between  the  molecules  of  a 
liquid  manifests  itself  near  the  surface  where  the  molecules  are 
subject  to  an  unbalanced  internal  force.  The  condition  of  a 
liquid  near  its  surface  is  roughly  depicted  in  Fig.  15,  where  the 
dots  A,  B,  and  C  represent  molecules  and  the  circles  represent  the 


© 


Fig.  15 


spheres  within  which  lie  all  of  the  other  molecules  which  exert  an 
appreciable  attraction  upon  A,  B,  and  C.  The  shaded  portions 
represent  those  molecules  whose  attractions  are  unbalanced. 
These  unbalanced  forces  will  evidently  tend  to  diminish  the  sur- 
face to  a  minimum  value.  That  is,  the  contraction  of  the  surface 
of  a  liquid  involves  the  expenditure  of  energy  by  the  liquid.  The 
surface  film  of  a  liquid  is  consequently  in  a  state  of  tension. 
Some  liquids  wet  the  walls  of  a  glass  capillary  tube  while  others 

do  not.  When  the  liquid 
wets  the  tube,  the  surface  is 
concave  and  the  liquid  rises 
in  the  tube ;  on  the  other 
hand,  when  the  liquid  does 
not  adhere,  the  surface  is 
convex  and  the  liquid  is 
depressed  in  the  tube.  The 
law  governing  the  elevation 
or  depression  of  a  liquid  in 
a  capillary  tube  was  discovered  by  Jurin  and  may  be  stated 
thus :  —  -  The  elevation  or  depression  of  a  liquid  in  a  capillary  I 
tube  is  inversely  proportional  to  the  diameter  of  the  tube.  Let  a  * 
capillary  tube  of  radius  r,  as  shown  in  Fig.  16,  be  immersed  in  a 
vessel  of  liquid  whose  density  is  d,  and  let  the  elevation  of  the 
liquid  in  the  capillary  be  denoted  by  h.  Then  the  weight  of  the 
column  of  liquid  in  the  capillary  will  be  irrzhdg,  where  g  is  the  ac- 
celeration due  to  gravity.  The  force  sustaining  this  weight  is 
the  vertical  component,  27,-rycosO,  of  the  force  due  to  the  tension 


Fig.  16 


LIQUIDS 


75 


of  the  liquid  surface  at  the  walls  of  the  tube,  7  being  the  surface 
tension  and  B  the  angle  of  contact  of  the  liquid  surface  with  the 
walls  of  the  tube. 
Therefore 

irr^hdg  =  2  irry  cos  0, 
or 


2cos0 

In  the  case  of  water  and  many  other  liquids  B  is  so  small  that  we 
may    write    6  =  0,   the   foregoing 
expression  becoming 


7  =  1/2  h  dgr. 


(24) 


Thus  the  surface  tension  of  a 
liquid  can  be  calculated  provided 
its  density  and  the  height  to  which 
it  rises  in  a  previously  calibrated 
tube  is  known.  When  h  and  r  are 
expressed  in  centimeters,  7  will  be 
expressed  in  dynes  per  centimeter 
or  ergs  per  square  centimeter. 

A  simple  form  of  apparatus  for 
the  determination  of  surface  tension 
used  by  the  author  is  shown  in 
Fig.  17.  A  capillary  tube,  A,  of 
uniform  bore  is  sealed  to  a  glass 
rod,  E,  which  is  held  in  position 
in  the  test  tube,  B,  by  means  of 
a  cork  stopper.  A  short  right-' 
angled  tube,  Z),  and  a  thermom- 
eter, F,  are  also  passed  through  the 
same  cork  stopper.  The  liquid 
whose  surface  tension  is  to  be 
measured  is  introduced  into  the 
tube,  B,  the  cork  inserted  and 
the  tube  placed  inside  of  the 


Fig.  17 


larger  tube,  C,  containing  a  liquid  of  known  boiling-point.  When 
the  thermometer,  F,  has  become  stationary,  the  capillary  elevation 
of  the  liquid  is  measured  with  a  cathetometer.  The  tube,  D, 
permits  the  escape  of  vapor  from  the  liquid  in  B  and  at  the 


76 


THEORETICAL  CHEMISTRY 


same  time  insures  equality  of  pressure  inside  and  outside  of 
the  apparatus.  The  spiral  tube,  G,  serves  as  an  air  condenser, 
preventing  loss  of  vapor  from  the  liquid  in  the  outer  tube.  The 
surface  tension  of  a  liquid  has  been  found  to  depend  upon  the 

nature  of  the  liquid  and  also  upon 
its  temperature. 

It  has  recently  been  shown  by 
Richards  and  Coombs*,  that  the 
values  of  the  surface  tension  of 
liquids  as  determined  by  the  cap- 
illary method,  are  in  error,  owing 
to  the  fact  that  the  values  of  h, 
in  equation  (24),  as  ordinarily 
measured,  are  slightly  less  than  the 
true  values  of  h,  when  measured 
from  a  strictly  horizontal  surface. 
To  avoid  this  source  of  error, 
Richards  and  Coombs  devised  the 
apparatus  shown  in  Fig.  18,  where 
the  larger  tube  has  a  sufficient 
diameter  (38  mm.)  to  secure  a  per- 
fectly flat  surface.  The  height  to 
which  a  liquid  rises  in  the  cap- 
illary tube  is  measured  by  means 
of  a  cathetometer:  the  difference 
between  the  reading  of  the  tele- 


Fig.  18 


scope  Iwhen  focussed,  (1)  on  the  capillary  meniscus  and,  (2)  on 
the  horizontal  surface  in  the  wide  tube,  is  the  true  capillary 
rise,  h. 

Surface  Tension  and  Molecular  Weight.  In  1886,  Eotvosf 
showed  that  the  surface  tension  multiplied  by  the  two-thirds 
power  of  the  molecular  weight  and  specific  volume  is  a  function 
of  the  absolute  temperature,  or 


where  7  is  the  surface  tension,  M  the  molecular  weight,  v  the  spe- 
cific volume  or  reciprocal  of  the  density,  and  T  the  absolute  tem- 


*  Jour.  Am.  Chem.  Soc.,  37,  1656  (1915). 
t  Wied.  Ann.,  27,  448  (1886). 


LIQUIDS  77 

perature.  Ramsay  and  Shields*  modified  the  equation  of  Eot- 
vos  as  follows:  — 

7  (Mv)  "  =  k  (tc  -  t  -  6),  (25) 

tc  being  the  critical  temperature  of  the  liquid,  t  the  temperature 
of  the  experiment,  and  k  a  constant  independent  of  the  nature  of 
the  liquid.  The  physical  significance  of  the  two-thirds  power  of 
the  molecular  volume  has  been  explained  by  Ostwald  in  the  follow- 
ing manner:  —  Assuming  the  molecules  to  be  spherical,  we  shall 
have  for  two  different  liquids,  the  proportion 

•Vi:  F2::n3:r23, 

where  Vi  and  F2  represent  the  volumes  and  r±  and  r2  the  radii  of 
their  respective  molecules.  Similarly  the  ratio  of  the  surfaces, 
Si  and  £2,  of  the  molecules  in  terms  of  their  respective  radii,  will 
be 

From  these  two  proportions  it  follows  that  the  ratio  of  the  molec- 
ular surfaces  in  terms  of  the  molecular  volumes,  will  be 

Si  :S2  ::  Vj  :  F2i 

Making  use  of  the  value  of  M  as  determined  in  the  gaseous  state, 
Ramsay  and  Shields  found  the  value  of  k  for  a  large  number  of 
liquids  to  be  equal  to  2.12  ergs.  Among  the  liquids  for  which 
this  value  of  k  was  found,  were  benzene,  carbon  tetrachloride, 
carbon  disulphide  and  phosphorus  trichloride.  For  certain  other 
liquids,  such  as  water,  methyl  and  ethyl  alcohols  and  acetic  acid, 
k  was  found  to  have  values  much  smaller  than  2.12.  Ramsay 
and  Shields  attributed  these  abnormalities  to  an  increase  in  molec- 
ular weight  due  to  association,  and  suggested  that  the  degree  of 
association  might  be  calculated  from  the  equation 

x*  =  2.12//c', 
or 

*=^f>  (26) 


where  x  denotes  the  factor  of  association,  and  kf  is  the  value  of 
the  constant  for  the  associated  liquid  in  equation  (25).  It  was 
further  pointed  out  by  Ramsay  and  Shields  that  equation  (25) 

*  Zeit.  phys.  Chem.,  12,  431  (1893). 


78  THEORETICAL  CHEMISTRY 

affords  a  means  of  calculating  the  molecular  weight  of  a  pure 
liquid,  provided  we  assume  that  the  mean  value  of  k  is  2.12  for 
a  non-associated  liquid.  Since  it  is  not  an  easy  matter  to  deter- 
mine the  critical  temperature  with  accuracy,  Ramsay  and  Shields 
made  use  of  a  differential  method,  and  thus  eliminated  tc  from 
equation  (25).  If  the  surface  tension  of  a  liquid  be  measured  at 
two  temperatures  ti  and  fe,  and  the  corresponding  densities  are  di 
and  d-2,  we  shall  have 

7i(MM)i  =  fc(k-*i-6),  (27) 

and 

=  fcft-fe-6).  (28) 


Subtracting  equation  (28)  from  equation  (27),  we  obtain 

T.W*)»-pW*)»_fc..a.iat  (29) 

£2  —  ti 

or  solving  equation  (29)  for  M  ,  we  have 
'kdj-dj-  (t2  - 


M 


The  method  of  Ramsay  and  Shields  is  the  best  known  method  for 
the  determination  of  the  molecular  weight  of  a  pure  liquid. 

If  M  is  known  to  be  the  same  in  the  liquid  and  gaseous  states, 
or  in  other  words,  if  k  is  independent  of  the  temperature,  even 
though  its  value  is  not  exactly  2.12,  the  critical  temperature  of 
the  liquid  can  be  calculated  by  means  of  equation  (25).  In  order 
that  the  correct  value  of  the  critical  temperature  may  be  obtained, 
Ramsay  and  Shields  found  it  necessary  to  use  the  specific  value 
of  k  for  the  liquid  whose  critical  temperature  is  sought.  As  an 
illustration  of  the  method  of  calculation,  the  following  example  is 
taken  from  the  work  of  Ramsay  and  Shields. 

For  carbon  disulphide, 

7  at  19°.4  =  33.58  7  at  46°.l  =  29.41 

d  at  19°.4  =  1.264  d  at  46°.l  =  1.223. 

We  have  then  for  7  (M/d)l,  at  the  two  temperatures, 

(76/1.264)!  X  33.58  =  515.4, 
and 

(76/1.223)1  X  29.41  =  461.4. 


LIQUIDS  79 

Substituting  in  the  equation 


71  -  72  _ 

fe  -  fe  ' 

we  have, 

8ff  "  to1/  -  2.022. 
46.1  —  19.4 

This  value  of  k  is  so  nearly  equal  to  the  mean  value,  2.12,  that  we 
assume  M  to  be  the  same  in  the  liquid  and  gaseous  states,  and 
therefore  we  may  substitute  in  equation  (1)  and  calculate  the 
critical  temperature  of  carbon  disulphide  thus, 

7  (M/d)*  =  k(tc-t-  6), 
or  solving  for  tc,  we  have 

7 


Substituting  the  data  given  above,  in  the  preceding  equation,  we 
obtain 

tc  =  515.4/2.022  +  6  +  19.4, 
or  tc  =  280°.3  C. 

Surface  Tension  and  Drop-Weight.  Morgan  and  his  co- 
workers,*  from  measurements  of  the  volumes  of  a  single  drop 
falling  from  the  carefully-ground  tip  of  a  capillary  tube,  have 
shown  that  the  weight  of  the  falling  drop  from  such  a  tip  can  be 
used  in  place  of  the  surface  tension  in  the  equation  of  Ramsay 
and  Shields  for  the  calculation  of  molecular  weights  and  critical 
temperatures.  The  modified  equation  may  be  written  thus :  — 

wi  (Mfdifi  —  w2  (M/d*)*       7  /Qn\ 

— — =  K,  (60) 

where  wi  and  w2  are  the  respective  weights  of  the  falling  drop 
at  the  temperatures,  ti  and  fe.  The  value  of  k  obviously  depends 
upon  the  tip  employed. 

The  results  obtained  by  the  drop-weight  method  have  been 
shown  to  be  more  trustworthy  than  those  obtained  by  the  method 
of  capillary  elevation.  Morgan  has  further  pointed  out  that 

*  Jour.  Am.  Chem.  Soc.,  30,  360  (1908);  30,  1055  (1908).  See  also  papers 
by  Harkins  (Jour.  Am.  Chem.  Soc.  39,  572  (1917)  and  Lohnstein  Z.  phys. 
Chem.  84,  410  (1913). 


80  THEORETICAL  CHEMISTRY 

when  the  epxerimental  data  are  substituted  in  the  preceding 
formula,  the  magnification  of  the  experimental  errors  is  appreci- 
ably greater  than  when  use  is  made  of  the  original  formula, 

w  (M/d)*  =  k(tc-t-  6).  (31) 

Morgan  recommends  therefore  that  this  formula  be  used  for 
the  determination  of  molecular  weights.  After  having  calibrated 
a  particular  tip  with  pure  benzene  (a  liquid  which  is  known  to  be 
non-associated),  and  thus  ascertaining  the  value  of  k,  the  drop- 
weights  at  several  different  temperatures  are  determined.  If 
we  assume  M  to  have  the  same  value  in  the. liquid  and  gaseous 
states,  the  value  of  tc  can  be  computed  by  substituting  the  experi- 
mental data  in  the  preceding  equation.  If  at  the  different  tem- 
peratures at  which  drop- weights  are  determined,  the  same  value 
of  tc  is  obtained,  then  we  may  infer  that  the  liquid  is  non-asso- 
ciated and,  therefore,  that  the  assumption  made  as  to  the  value  of 
M  is  confirmed.  It  is  a  singular  fact  that  the  calculated  value 
of  tc  for  some  liquids  does  not  agree  with  the  experimental  value, 
although  it  remains  constant  throughout  an  extended  range  of 
temperatures.  Morgan  considers  a  constant  value  of  tc  to  be  an 
indication  of  non-association,  even  if  the  value  is  fictitious.  In 
this  method  the  constancy  of  the  calculated  value  of  the  critical 
temperature  becomes  the  criterion  of  molecular  association  and 
thus  affords  a  means  of  determining  whether  the  molecular  weight 
in  the  liquid  state  is  identical  with  that  in  the  gaseous  state.  The 
values  of  tc  calculated  from  the  drop-weights  of  an  associated 
liquid  become  steadily  smaller  as  the  temperature  increases.  A 
large  number  of  liquids  have  been  studied  by  this  method,  and 
the  results  indicate  that  many  of  the  substances  which  were  con- 
sidered to  be  associated  by  Ramsay  and  Shields  are  in  reality 
non-associated;  in  fact,  it  appears  from  the  work  of  Morgan  that 
association  is  much  less  common  among  liquids  than  has  hitherto 
been  supposed. 

Viscosity.  The  resistance  experienced  by  one  portion  of  a 
liquid  in  moving  over  another  portion  is  called  viscosity.  The 
viscosity  of  liquids  varies  greatly,  some  liquids,  such  as  ether,  being 
very  mobile,  while  others,  such  as  tar,  are  extremely  viscous.  It- 
has  been  shown  experimentally  that  the  tangential  force,  /,  re- 
quired to  maintain  a  constant  difference  between  the  velocities 
of  two  parallel  layers  of  liquid  moving  in  the  same  direction  varies 


LIQUIDS  81 

directly  with  the  difference  in  velocity,  u,  and  the  area,  A,  of  the 
surface  of  contact  of  the  two  layers,  and  inversely  as  the  distance, 
d,  between  the  layers.  That  is, 


where  rj,  is  a  proportionality  factor  known  as  the  coefficient  of 
viscosity.  The  unit  of  viscosity  is  the  poise:  this  may  be  defined 
as  the  force  necessary  to  cause  two  parallel  liquid  surfaces  of  unit 
area  and  unit  distance  apart  to  slide  past  one  another  with  unit 
velocity.  The  viscosity  of  a  liquid  is  generally  measured  by  ob- 
serving the  time  required  for  a  definite  volume  of  liquid  to  flow 
through  a  standardized  capillary  tube  under  a  known  difference 
of  pressure.  The  law  governing  the  flow  of  liquids  through  cap- 
illary tubes  was  discovered  by  Poiseuille*  and  may  be  expressed 
by  the  equation 


in  which  V  denotes  the  volume  of  liquid  of  viscosity  tj,  flowing 
through  a  capillary  tube  of  length  /,  and  radius  r,  in  the  time  t}  and 
under  the  pressure  P.  If  the  times  of  flow  of  equal  volumes 
of  two  liquids  through  the  same  capillary  are  measured  under  the 
same  driving  pressure,  it  follows  from  equation  (33)  that 

^  =  ^,  (34) 

172      d2tz      + 

where  iji  and  172  denote  the  coefficients  of  viscosity  of  the  two 
liquids,  di  and  dz  their  densities  and  t\  and  t2  their  times  of  flow. 
If  a  liquid  of  known  viscosity  be  selected  as  a  standard,  equation 
(34)  may  be  used  to  calculate  the  so-called  "  relative  viscosity  "  of 
other  liquids.  Water  is  quite  generally  accepted  as  the  standard 
of  reference  in  determinations  of  relative  viscosity.  In  calculat- 
ing the  relative  viscosity  of  a  liquid  referred  to  water,  therefore, 
we  employ  the  following  equation* 


in  which  171,  d\  and  ti  denote  the  viscosity,  density  and  time  of 
flow,  respectively,  of  the  liquid  under  investigation,  and  dw  and  tw 

*  Ann.  Chim.  Phys.,  37,  50  (1843). 


82 


THEORETICAL  CHEMISTRY 


denote  the  density  and  time  of  flow  of  water.  Relative  viscosities 
may  be  transformed  into  absolute  viscosities  by  multiplying  the 
relative  viscosity  by  the  absolute  viscosity  of  water  at  the  same 
temperature  as  that  at  which  the  measurements  are  made. 

A  convenient  type  of  viscometer  is  shown  in  Fig.  19.  The  liquid 
is  allowed  to  flow  under  its  own  pressure  through  the  capillary 
db.  An  accurately  measured  volume  of  the  liquid  is  introduced 
at  /,  and  by  applying  suction  at  a  it  is  drawn  up  the  tube  until 
the  liquid  has  risen  above  the  mark  c.  The  liquid  is  now  allowed 
to  flow  down  the  tube,  and  the  time  required  for 
the  meniscus  to  move  from  the  mark  c,  to  the  lower 
mark  d,  is  measured  with  a  stop-watch.  The  exper- 
iment is  now  repeated  with  water,  after  which  the 
viscosity  of  the  liquid  can  be  calculated  by  means 
of  equation  (35). 

The  reciprocal  of  viscosity  is  called  fluidity,  i.e., 
<t>  =  I/??.  Fluidity  may  be  regarded  as  a  measure 
of  the  tendency  of  a  liquid  to  flow,  whereas  viscosity 
is  a  measure  of  the  resistance  which  a  liquid  offers 
to  flowing. 

The  viscosity  of  liquids  has  been  found  to  decrease 
about  two  per  cent  for  each  degree  of  rise  in  tem- 
perature. Increase  of  pressure,  on  the  other  hand, 
causes  the  viscosity  of  liquids  to  increase,  the  change 
in  viscosity  corresponding  to  equal  increments  of 
pressure  being  greater  at  high  pressures  than  at  low. 

While  there  appears  to  be  an  abundance  of  experimental  evi- 
dence to  prove  that  the  viscosity  of  a  mixture  of  two  or  more 
liquids  is  not  equal  to  the  sum  of  the  viscosities  of  the  compo- 
nents, it  is  claimed  by  Bingham*  that  there  is  both  theoretical  and 
experimental  justification  for  the  statement  that  fluidities  are 
additive.  He  points  out  that  the  fluidity  0,  of  a  mixture  of  two 
liquids  whose  fluidities  are  0i,  and  02,  respectively,  may  be  repre- 
sented by  the  equation 

<t>  =  a0i  +  602,  (36) 

where  a  and  b  denote  the  volume  concentrations  of  the  two  liquids. 
This  equation  has  been  found  to  hold  in  the  case  of  liquids  which 

*  Bingham  has  pointed  out  that  a  viscous  liquid  will  flow  under  an  in- 
finitesimal pressure,  whereas  a  plastic  substance  will  not  flow  until  the  pres- 
sure has  exceeded  a  definite  value,  termed  the  "yield  point." 


Fig.  19 


LIQUIDS 


83 


are  "  inert,"  i.e.,  liquids  which  are  not  only  individually  non- 
associated  but  also  exhibit  no  tendency  to  combine  with  each 
other. 

REFERENCES 

Stoichiometry.     Sydney  Young. 

The  Viscosity  of  Liquids.     Dunstan  and  Thole. 

Fluidity  and  Plasticity.     Bingham. 

PROBLEMS 

1.  The  density  of  nitric  oxide  at  0D  and  760  mm.  is  1.3402;  the  critical 
temperature  and  the  critical  pressure  are  179.5°  abs.  and  71.2  atmos- 
pheres respectively.     Calculate  the  atomic  weight  of  nitrogen  by  Ber- 
thelot's  equation. 

2.  The  weight  of  1  liter  of  methyl  fluoride,  CH3F,  at  three  different 
pressures  is  as  follows: 

1.54507  at  760  mm., 
1.53576  at  506.67  mm., 
1.52665  at  253.33  mm. 

Calculate  the  atomic  weight  of  fluorine  by  the  method  of  limiting  den- 
sities. 

3.  The  critical  temperature  of  normal  pentane  is  197.15°.    The  ortho- 
baric  densities  of  normal  pentane  in  the  liquid  and  vapor  states  at  several 
temperatures  are  as  follows: 


Temp. 

di 

dv 

110° 

0.5248 

0.0203 

130 

0.4957 

0.0310 

150 

0.4604 

0.0476 

180 

0.3867 

0.0935 

190 

0.3445 

0.1269 

195 

0.3065 

0.1609 

From  the  foregoing  data  calculate  the  critical  density  of  normal  pentane. 

4.  A  volume  of  50  liters  of  air  in  passing  through  a  liquid  at  22°  C. 
causes  the  evaporation  of  5  grams  of  substance,  the  molecular  weight  of 
which  is  100.    What  is  the  vapor  pressure  of  the  liquid  in  grams  per 
square  centimeter?  Ans.  25. 

5.  The  boiling-point  of  ethyl  propionate  is  98°. 7  C.  and  its  heat  of 
vaporization  is  77.1  calories.     Calculate  its  molecular  weight. 

6.  The  heat  of  vaporization  of  liquid  ammonia  at  its  boiling-point, 
under  atmospheric  pressure  (—33°. 5  C.)  is  341  calories.     Is  liquid  am- 
monia associated? 


84  THEORETICAL  CHEMISTRY 

7.  Calculate  the  surface  tension  of  benzene  in  dynes  per  centimeter, 
the  radius  of  the  capillary  tube  being  0.01843  cm.,  the  density  of  the  liquid, 
0.85,  and  the  height  to  which  it  rises  in  the  capillary,  3.213  cm. 

Ans.  24.71  dynes/cm. 

8.  Find  the  molecular  weight  of  benzene,  the  surface  tension  at  46°  C. 
being  24.71   dynes  per  centimeter,  its  critical  temperature,  288°.5  C., 
its  density,  0.85  and  the  value  of  k  =  2.12.  Ans.  77.7 

9.  At  14°.8  C.  acetyl  chloride  (density  =  1.124)  ascends  to  a  height 
of  3.28  cm.  in  a  capillary  tube  the  radius  of  which  is  0.01425  cm.    At 
46°. 2  C.  in  the  same  tube  the  elevation  is  2.85  cm.  and  the  density  = 
1.064.    Calculate  the  critical  temperature  of  acetyl  chloride. 

Ans.  234.6°. 

10.  From  a  certain  tip  the  weights  of  a  falling  drop  of  benzene  are 
35.329  milligrams   (temp.  =  11°.4,   density  =  0.888)   and   26.530   milli- 
grams (temp.  =  68°.5,  density  =  0.827).     The  molecular  weight  is  the 
same  in  the  liquid  and  gaseous  states.     Calculate  the  critical  temper- 
ature of  benzene.  Ans.  286.1°. 


r 

CHAPTER  IV 
SOLIDS 

General  Properties  of  Solids.  Solids  differ  from  gases  and 
liquids  in  possessing  definite,  individual  forms.  Matter  in  the 
solid  state  is  capable  of  resisting  considerable  shearing  and  tensile 
stresses.  In  terms  of  the  kinetic  theory  of  matter,  the  mutual 
attractive  forces  exerted  by  the  molecules  of  solids  must  be  re- 
garded as  superior  to  the  attractive  forces  between  the  molecules 
of  gases  and  liquids.  With  one  or  two  exceptions,  all  solids  ex- 
pand when  heated,  but  there  is  no  simple  law  expressing  the 
relation  between  the  increment  of  volume  and  the  temperature. 
Rigidity  is  another  characteristic  property  of  solids,  it  being  much 
more  apparent  in  some  than  in  others.  Many  solids  are  constantly 
undergoing  a  process  of  transformation  into  the  gaseous  state  at 
their  free  surfaces,  such  a  change  being  known  as  sublimation. 
Just  as  a  gas,  when  sufficiently  cooled  passes  into  the  liquid 
state,  so  on  cooling  a  liquid  below  a  certain  temperature,  it  passes 
into  the  solid  state.  The  reverse  transformations  are  also  possible, 
a  solid  being  liquefied  when  sufficiently  heated,  and  the  resulting 
liquid  completely  vaporized  if  the  heating  be  continued.  Heat 
energy  is  required  to  effect  transition  from  the  solid  to  the  liquid 
state,  just  as  heat  energy  is  required  to  effect  transition  from  the 
liquid  to  the  gaseous  state. 

Obviously  a  substance  in  the  solid  state  contains  less  energy 
than  it  does  in  the  liquid  state.  The  number  of  calories  required 
to  melt  1  gram  of  a  solid  substance  is  called  its  heat  of  fusion. 
It  is  often  difficult  to  decide  whether  a  substance  should  be  classi- 
fied as  a  solid  or  as  a  liquid.  For  example  the  behavior  of  certain 
amorphous  substances,  such  as  pitch,  amber  and  glass,  is  similar 
to  that  of  a  very  viscous,  inelastic  liquid.  Solids  are  generally 
classified  as  crystalline  or  amorphous.  In  crystalline  solids  the 
molecules  are  supposed  to  be  arranged  in  some  definite  order,  this 
arrangement  manifesting  itself  in  the  crystal  form.  An  amor- 
phous solid  on  the  other  hand  may  be  considered  as  a  liquid 

85 


86  THEORETICAL  CHEMISTRY 

possessing  great  viscosity  and  small  elasticity.  The  physical 
properties  of  amorphous  solids  have  the  same  values  in  all  direc- 
tions, whereas  in  crystalline  solids  the  values  of  these  properties 
may  be  different  in  different  directions.  When  an  amorphous 
solid  is  heated,  it  gradually  softens  and  eventually  acquires  the 
properties  characteristic  of  a  liquid,  but  during  the  process  of  heat- 
ing there  is  no  definite  point  of  transition  from  the  solid  to  the 
liquid  state.  On  the  other  hand,  when  a  crystalline  solid  is 
heated  there  is  a  sharp  change  from  one  state  to  the  other  at  a 
definite  temperature,  this  temperature  being  termed  the  melting- 
point. 

Crystallography.  The  study  of  the  definite  geometrical  forms 
assumed  by  crystalline  solids  is  termed  crystallography.  The 
number  of  crystalline  forms  known  is  exceedingly  large,  but  it  is 
possible  to  reduce  the  many  varieties  to  a  few  classes  or  systems 
by  referring  their  principal  elements  —  the  planes  —  to  definite 
lines  called  axes.  These  axes  are  so  drawn  within  the  crystal  that 
the  crystal  surfaces  are  symmetrically  arranged  about  them. 
This  system  of  classification  was  proposed  by  Weiss  in  1809. 
He  showed  that,  notwithstanding  the  multiplicity  of  crystal  forms 
encountered  in  nature,  it  is  possible  to  consider  them  as  belonging 
to  one  of  six  systems  of  crystallization. 

The  six  systems  of  Weiss  are  as  follows :  — 

1.  The  Regular  System.     Three  axes  of  equal  length,  inter- 
secting each  other  at  right  angles  (Fig.  20). 

2.  The  Tetragonal  System.     Two  axes  of  equal  length  and  the 
third  axis  either  longer  or  shorter,  all  three  axes  intersecting  at 
right  angles  (Fig.  21). 

3.  The  Hexagonal  System.     Three  axes  of  equal  length,  all  in 
the  same  plane  and  intersecting  at  angles  of  60°,  and  a  fourth  axis, 
either  longer  or  shorter  and  perpendicular  to  the  plane  of  the 
other  three  (Fig.  22). 

4.  The  Rhombic  System.     Three  axes  of  unequal  length,  all 
intersecting  each  other  at  right  angles  (Fig.  23). 

5.  The  Monoclinic  System.     Three  axes  of  unequal  length,  two 
of  which  intersect  at  right  angles,  while  the  third  axis  is  per- 
pendicular to  one  and  not  to  the  other  (Fig.  24). 

6.  The  Triclinic  System.     Three  axes  of  unequal  length  no  two 
of  which  intersect  at  right  angles  (Fig.  25). 


SOLIDS 


87 


The  position  of  a  plane  in  space  is  determined  by  three  points 
in  a  system  of  coordinates,  and  consequently  the  position  of  the 
face  of  a  crystal  is  likewise  determined  by  its  points  of  intersection 
with  the  three  axes,  or  by  the  distances  from  the  origin  of  the 
system  of  coordinates  at  which  the  plane  of  the  crystal  face 
intersects  the  three  axes.  These  distances  are  called  the  para- 
meters of  the  plane. 


i 
Fig.  20 


o 

Fig.  21 


Fig.  23 


Fig.  24 


Fig.  25 


The  fundamental  law  of  crystallography  discovered  by  Steno 
in  1669  may  be  stated  thus :  —  The  angle  between  two  given  crystal 
faces  is  always  the  same  for  the  same  substance.  The  fact  that  every 
crystalline  substance  is  characterized  by  a  constant  interfacial 
angle,  affords  a  valuable  means  of  identification  which  is  used 
by  both  chemists  and  mineralogists.  The  instrument  employed 
for  the  measurement  of  the  interfacial  angles  of  crystals  is  called 
a  goniometer.  The  crystal  to  be  measured  is  mounted  at  the 
center  of  the  graduated  circular  table  of  the  goniometer,  and 
the  image  of  an  illuminated  slit,  reflected  from  one  surface  of  the 
crystal,  is  brought  into  coincidence  with  the  cross-wires  in  the 


88  THEORETICAL  CHEMISTRY 

eye-piece  of  the  telescope.  The  table  is  then  turned  until  the 
image  of  the  slit,  reflected  from  the  adjacent  face  of  the  crystal, 
coincides  with  the  cross-wires.  The  interfacial  angle  of  the 
crystal  is  determined  by  the  number  of  degrees  through  which 
the  table  has  been  turned. 

Properties  of  Crystals.  The  properties  of  all  crystals,  except 
those  belonging  to  the  regular  system,  exhibit  differences  depend- 
ent upon  the  direction  in  which  the  particular  measurements 
are  made.  Thus  the  elasticity,  the  thermal  and  electrical  con- 
ductivities, and  in  fact  all  of  the  physical  properties  of  crystals 
which  do  not  belong  to  the  regular  system,  have  different  values 
in  different  directions.  Crystals  whose  physical  properties  have 
the  same  values  in  all  directions  are  termed  isotropic,  while  those 
in  which  the  values  are  dependent  upon  the  direction  in  which 
the  measurements  are  made,  are  called  anisotropic.  Certain  amor- 
phous substances,  such  as  glass,  which  are  normally  isotropic,  may 
become  anisotropic  when  subjected  to  tension  or  compression. 
The  phenomenon  of  double  refraction  observed  in  all  crystals,  ex- 
cept those  belonging  to  the  regular  system,  is  due  to  their  aniso- 
tropic character.  Crystals  belonging  to  the  tetragonal  and  hex- 
agonal systems  resemble  each  other  in  one  respect,  viz. :  that  in 
all  of  them  there  is  one  direction,  called  the  optic  axis,  (coincident 
with  the  principal  crystallographic  axis) ,  along  which  a  ray  of  light 
is  singly  refracted,  while  in  all  other  directions  it  is  doubly  refracted. 
In  crystals  belonging  to  the  rhombic,  monoclinic,  and  triclinic 
systems,  there  are  always  two  directions  along  which  a  ray  of 
light  is  singly  refracted.  A  crystal  of  Iceland  spar  (CaCO3) 
affords  a  beautiful  illustration  of  double  refraction.  On  placing 
a  rhomb  of  this  substance  over  a  piece  of  white  paper  on  which 
there  is  an  ink  spot,  two  spots  will  be  seen.  On  turning  the  crys- 
tal, one  spot  will  remain  stationary  while  the  other  spot  will 
revolve  about  it.  This  property  of  Iceland  spar  is  utilized  in  the 
construction  of  Nicol  prisms  for  polariscopes. 

The  examination  of  sections  of  anisotropic  crystals  in  a  polari- 
scope  between  crossed  Nicol  prisms,  reveals  something  as  to  their 
crystal  form.  As  has  been  stated,  crystals  of  the  tetragonal  and 
hexagonal  systems  are  uniaxial.  If  a  section  is  cut  from  such  a 
crystal  perpendicular  to  the  optic  axis,  and  this  is  placed  between 
the  crossed  Nicol  prisms  of  a  polariscope,  in  a  convergent  beam 
of  white  light,  a  dark  cross  and  concentric,  spectral-colored  circles 


SOLIDS  89 

will  be  observed,  Fig.  26.  Upon  turning  the  analyzer  through 
90°  the  colors  of  the  circles  will  change  to  the  respective  comple- 
mentary colors  and  the  dark  cross  will  become  light.  Crystals 
of  the  rhombic,  monoclinic,  and  triclinic  systems  are  biaxial.  If 
a  section  of  a  biaxial  crystal,  cut  perpendicular  to  the  line  bisect- 
ing the  angle  between  the  tv/o  axes,  be  placed  in  the  polariscope 
and  examined  as  in  the  preceding  case,  a  series  of  concentric 
spectral-colored  lemniscates  surrounding  two  dark  centers  and 


Fig.  26  Fig.  27 

pierced  by  dark,  hyperbolic  brushes,  will  be  observed,  as  shown 
in  Fig.  27.  On  rotating  the  analyzer,  the  colors  will  change  to  the 
corresponding  complementary  colors,  as  in  the  case  of  uniaxial 
crystals.  The  appearance  of  these  figures  is  so  varied  and  char- 
acteristic as  to  furnish,  in  many  cases,  a  very  satisfactory  means 
of  identifying  anisotropic  crystals. 

Etch  Figures.  The  solubility  of  crystals  has  been  shown  to 
be  different  in  different  directions.  Thus,  if  the  surface  of  a  crys- 
talline substance  be  highly  polished  and  then  treated  for  a  short 
time  with  a  suitable  solvent,  faint  patterns,  known  as  etch  figures, 
will  appear  as  a  result  of  the  inequality  of  the  rate  of  solution  in 
different  directions.  When  these  figures  are  examined  under  the 
microscope  the  crystal-form  can  generally  be  determined.  The 
examination  of  etch  figures  has  come  to  be  of  prime  importance 
to  the  metallographer.  Thus,  when  an  appropriate  solvent  is 
applied  to  the  polished  surface  of  an  alloy,  not  only  is  the  crystal 
form  revealed  by  the  etch  figures,  but  also  the  presence  of  various 
chemical  compounds  may  be  recognized.  By  a  careful  study 
of  the  etch  figures  developed  on  the  surface  of  highly  polished 
steel,  the  metallographer  may  gather  important  information  as  to 
its  previous  history,  especially  its  heat  treatment. 

Crystal  Form  and  Chemical  Composition.  From  the  pre- 
ceding paragraphs  it  might  be  inferred  that  the  same  substance 
always  assumes  the  same  crystal  form.  While  this  is  true  in 


90  THEORETICAL  CHEMISTRY 

general,  there  are  some  substances  which  appear  in  several  dif- 
ferent crystal  forms.     This  phenomenon  is  termed  polymorphism. 

Calcium  carbonate  is  an  example  of  a  substance  crystallizing  in 
more  than  one  form.  As  calcite,  it  crystallizes  in  the  hexagonal 
system,  while  as  aragonite,  it  crystallizes  in  the  rhombic  system. 
Such  a  substance  is  said  to  be  dimorphous.  Of  the  several  factors 
controlling  polymorphism,  temperature  is  the  most  important. 
Thus  sulphur  crystallizes  at  temperatures  above  95°. 6  in  the 
monoclinic  system,  while  at  lower  temperatures  it  assumes  the 
rhombic  form.  The  temperature  at  which  it  changes  from  one 
form  into  the  other  is  termed  its  transition  temperature.  As  has 
been  mentioned  in  an  earlier  chapter  (p.  13),  some  substances 
may  crystallize  in  the  same  form,  the  characteristic  interfacial 
angles  being  nearly  identical.  Such  substances  are  said  to  be  iso- 
morphous.  This  phenomenon,  discovered  by  Mitscherlich,  has 
been  of  great  use  in  connection  with  the  earlier  investigations  on 
atomic  weights,  as  has  already  been  pointed  out. 

There  can  be  little  doubt  as  to  the  existence  of  an  intimate 
connection  between  crystalline  form  and  chemical  composition 
Ever  since  the  early  part  of  the  nineteenth  century,  when  Haiiy 
established  the  science  of  crystallography,  various  attempts  have 
been  made  by  chemists  and  crystallographers  to  connect  crystal- 
line form  with  chemical  constitution.  In  1906,  Barlow  and  Pope* 
made  a  most  notable  contribution  to  the  theories  concerning  the 
relation  between  crystalline  form  and  chemical  constitution. 
Their  ideas  may  be  summarized  as  follows :  —  If  each  atom  be 
considered  as  appropriating  a  certain  space,  called  its  sphere  of 
atomic  influence,  then  (l)~f  he  spheres  of  atomic  influence  are  so 
arranged  as  to  occupy  the  smallest  possible  volume  in  every  crystal; 
(2)  the  volumes  of  the  spheres  of  atomic  influence  in  any  substance 
are  proportional  to  the  valences  of  the  constituent  atoms;  (3)  the 
volumes  of  the  spheres  of  influence  of  the  atoms  of  different  elements 
of  the  same  valence  are  nearly  equal,  any  variation  being  in  har- 
mony with  their  relations  in  the  periodic  system.  Barlow  and  Pope 
have  shown  that  the  general  agreement  between  theory  and 
observation  is  most  satisfactory,  a  particularly  strong  argument 
in  favor  of  this  theory  is  the  very  plausible  explanation  which 
it  furnishes  for  a  large  number  of  crystallographic  facts. 

*  Jour.  Chem.  Soc.,  91,  1150  (1907). 


SOLIDS  91 

Compressibilities  of  the  Solid  Elements.  A  series  of  careful 
measurements  of  the  compressibilities  of  the  elements  by  T.  W. 
Richards  and  his  collaborators,*  has  revealed  the  fact  that  com- 
pressibility is  a  periodic  function  of  atomic  weight.  Richards  has 
advanced  some  interesting  suggestions  as  to  the  importance  of 
compressibility  in  connection  with  intermolecular  cohesion  and 
atomic  volume.  In  fact,  Richards'  theory  of  compressible  atoms 
may  be  regarded  as  a  valuable  supplement  to  the  theory  of  Barlow 
and  Pope  and,  taken  together,  these  two  theories  constitute  a 
rational  basis  for  the  science  of  chemical  crystallography. 

X-Rays  and  Crystal  Structure.  Much  light  has  recently  been 
shed  upon  the  question  of  the  arrangement  of  the  atoms  and 
molecules  within  crystals  by  the  interesting  investigations  which 
have  been  carried  out  on  the  transmission  and  reflection  of  X-rays 
by  crystalline  substances  It  is  now  a  well-established  fact  that 
X-rays  are  in  reality  light  waves  of  extremely  short  wave-length. 
The  wave-length  of  the  X-ray  is  approximately  1  X  10~8  cm.  or, 
in  other  words  the  wave-length  of  the  X-ray  is  of  the  same  order  of 
magnitude  as  the  distance  between  contiguous  atoms  in  a  crystal. 

In  1912,  Laue  pointed  out  that  the  regularly  arranged  atoms  of 
a  crystal  should  act  as  a  three-dimensional  diffraction  grating 
toward  X  rays.  He  showed  mathematically  that  on  traversing 
a  thin  section  of  a  crystal,  a  pencil  of  X-rays  should  give  rise  to 
a  diffraction  pattern  arranged  symmetrically  round  the  primary 
beam  as  a  center.  A  photographic  plate  placed  perpendicular 
to  the  path  of  the  rays  and  behind  the  crystal  should  reveal,  on 
development,  a  central  spot  due  to  the  action  of  the  primary  rays, 
and  a  series  of  symmetrically  grouped  spots  due  to  the  diffracted 
rays. 

Laue's  predictions  were  verified  experimentally  by  Friedrich 
and  Knipping,f  who  obtained  numerous  plates  showing  a  va- 
riety of  geometrical  patterns  corresponding  to  the  structural  dif- 
ferences of  the  crystals  examined.  The  analysis  of  the  Laue 
diffraction  patterns,  while  furnishing  valuable  information  as  to 
the  internal  structure  of  crystals,  is  nevertheless  extremely  com- 
plex. In  addition  to  the  original  method  of  Laue  for  the  in- 
vestigation of  crystal  structure  two  other  methods  have  been 
developed  which  have  proven  to  be  of  great  value. 

*  Zeit.  phys.  Chem.,  61,  77,  100,  171,  183  (1908). 
t  Ann.  d.  Phys.,  41,  971  (1913). 


92 


THEORETICAL  CHEMISTRY 


W.  H.  Bragg  and  his  son,  W.  L.  Bragg,*  pointed  out  that  the 
diffraction  of  X-rays  by  the  atoms  of  a  crystal  can  be  studied  much 
more  simply  by  employing  the  crystal  as  a  reflection  grating,  rather 
than  as  a  transmission  grating,  as  in  the  method  of  Laue.  The 
reflection  of  X-rays  from  the  face  of  a  crystal  may  be  likened  to 
the  reflection  of  a  beam  of  light  from  a  bundle  of  thin  glass  plates 
of  equal  thickness.  It  is  a  well-known  fact  that  reflection  from 
a  bundle  of  plates  will  take  place  only  at  definite  angles,  the 
values  of  which  will  be  dependent  upon  the  wave-length  of  the 

incident  light,  as  well  as  upon 
the  thickness  of  the  plates.  The 
plates  of  glass  correspond  to  the 
planes  of  atoms  within  the  crys- 
tal. In  order  to  study  the  reflec- 
tion of  X-rays  from  the  faces  of 
crystals,  the  Braggs  devised  an 
instrument  known  as  the  X-ray 
spectrometer,  Fig.  28.  In  this 
instrument,  a  beam  of  X-rays, 
preferably  of  a  single  wave-length, 
is  rendered  parallel  by  means  of 
two  narrow  slits  A  and  B;  after 
passing  through  these  slits,  the 
rays  are  reflected  from  one  of  the 
faces  of  a  crystal  C,  which  is 


Fig.  28 


mounted  as  a  reflection  grating  upon  the  table  of  the  spectrometer, 
and  the  reflected  beam  then  enters  the  so-called  "  ionization  cham- 
ber "  where  the  intensity  of  the  beam  is  measured.  In  place  of  the 
ionization  chamber  a  photographic  plate  may  be  used  but  the 
results  are  not  as  satisfactory.  The  intensities  and  positions  of 
the  various  lines  of  the  X-ray  spectra  obtained  by  reflection  from 
the  different  faces  of  the  crystal  serve  to  determine  the  relative 
positions  of  the  atoms  within  the  crystal. 

In  the  third  method  of  investigating  crystal  structure  by  means 
of  X-rays,  the  crystalline  substance  is  employed  in  the  form  of 
powder.  It  has  been  shown  by  A.  W.  Hullf,  the  originator  of 
the  method,  that  if  a  parallel  beam  of  mono-chromatic  X-rays 
is  allowed  to  pass  through  a  finely  divided  crystalline  substance, 

*  "  X-Rays  and  Crystal  Structure." 

t  Phys.  Rev.  10,  661  (1917);  Jour.  Am.  Chem.  Soc.,  41,  1168  (1918). 


SOLIDS 


93 


a  characteristic  spectrum,  made  up  of  fine  lines,  will  be  produced 
upon  a  properly  placed  photographic  film.  These  lines  corre- 
spond to  the  reflections  from  the  different  faces  of  the  minute 
crystalline  particles  of  the  powdered  substance.  Not  only  does 


Fig.  29 

the  method  of  powders  furnish  essentially  the  same  information 
as  that  supplied  by  the  spectrometric  method  but  it  does  so  in  a 
single  experiment,  whereas,  with  the  spectrometric  method,  there 
must  be  as  many  separate  experiments  as  there  are  crystal  faces. 
Furthermore,  it  has  the  advantage  over  the  preceding  methods 


Fig.  30 

that  instead  of  a  well  developed  crystal  of  appreciable  size  merely 
a  very  small  amount  of  crystalline  powder  is  all  that  is  required; 
in  fact  Hull  has  found  that  the  amount  of  material  necessary  for 
a  complete  determination  of  crystal  structure  need  not  be  more 
than  one  cubic  millimeter.  In  the  diagram  of  Hull's  apparatus, 
shown  in  Fig.  29,  T'  is  a  transformer  for  operating  the  Coolidge 


94 


THEORETICAL  CHEMISTRY 


X-ray  tube,  x;  F  is  a  sheet  of  metal,  serving  as  a  filter;  Si  and 
S2  are  slits  in  thin  sheets  of  lead;  T  is  a  thin-walled  tube,  about 
one  mm.  in  diameter,  containing  the  powdered  crystal;  F  is  a 
narrow  strip  of  photographic  film  bent  over  a  semi-circular  strip 
of  brass,  concentric  with  T.  A  typical  spectrogram  is  reproduced 
in  Fig.  30. 

Application  of  the  foregoing  methods  has  revealed  the  crystal 
structure  of  a  large  number  of  familiar  minerals  and  chemical 
compounds.  For  example,  the  arrangement  of  the  atoms  in  a 
crystal  of  rock-salt  (NaCl)  has  been  found  to  be  similar  to  that 
shown  in  Fig.  31  A,  in  which  the  white  circles  represent  sodium 
atoms  and  the  black  circles,  chlorine  atoms.  Since  the  efficiency 

of  an  atom  as  a  reflector  of 
X-rays  is  proportional  to  its 
atomic  weight  it  follows  that 
the  relative  positions  of  the 
atoms  in  the  crystal  will  be 
revealed  by  the  relative 
intensities  of  the  reflected 
rays.  In  this  manner  the 
atoms  of  sodium  and  chlo- 
rine have  been  found  to 
alternate  as  shown  in  Fig. 
31  A.  Furthermore,  since  the 
wave-length  of  the  X-rays 
used  in  studying  the  crystal 
of  rock-salt  is  known,  the 
distance  between  the  planes 
of  the  atoms  can  be  cal- 


Had  NaCl  NaCl  NaC^UjIt 


JaCl 


Na  Cl  Na  Cl  N: 


Fig.  31 


culated;  this  distance  is  found  to  be  2.81  X  10~8  cm. 

The  arrangement  of  the  atoms  in  planes  is  shown  diagrammatic- 
ally  in  Fig.  31B.  One  of  the  most  striking  facts  which  is  revealed 
by  the  structure  of  this  and  other  crystals  is  the  absence  of  any 
molecular  grouping;  in  fact  the  crystal  itself  may  be  regarded  as 
a  magnified  molecule  held  together  by  interatomic  forces. 

In  like  manner,  X-ray  spectra  of  a  number  of  the  metals  have 
revealed  the  fact  that  in  the  majority  of  cases  the  arrangement 
of  the  atoms  is  such  as  might  be  expected  to  result  from  the  close 
packing  of  bodies  of  spherical  shape.  There  are  two  different 
ways  in  which  inelastic  spheres  of  equal  size  will  arrange  them- 


SOLIDS  95 

selves  when  closely  packed,  either  by  pressure  or  shaking.  The 
most  common  arrangement  is  that  which  is  known  as  the  "  face- 
centered  cubic  "  in  which  the  crystal  is  made  up  of  closely  packed 
cubes  having  an  atom  at  each  corner  and  also  at  the  center  of 
each  face.  The  other  grouping  is  the  "  body-centered  cubic  " 
in  which  an  atom  is  situated  at  the  corner  of  each  elementary 
cube  and  also  at  the  center:  the  body-centered  arrangement  is 
not  as  closely  packed  as  the  face-centered  arrangement  nor  is  it 
as  stable.  Among  the  metals  whose  atoms  have  the  face-centered 
arrangement  are  copper,  silver,  lead,  platinum  and  gold,  whereas 
sodium,  lithium,  iron,  chromium  and  tungsten  have  been  shown 
to  be  made  up  of  crystals  whose  atoms  have  the  body-centered 
cubic  arrangement. 

All  of  the  alkali  halides  as  well  as  the  divalent  metallic  oxides 
such  as  MgO,  CaO,  CdO,  and  NiO  resemble  sodium  chloride  in 
structure.  A  study  of  the  structure  of  different  crystalline  com- 
pounds has  led  to  the  conclusion,  that  no  direct  relation  exists 
between  the  arrangement  of  the  atoms  and  the  corresponding 
valences.  There  is  considerable  evidence,  however,  in  favor  of 
the  view  that  the  atoms  of  crystals  of  sodium  chloride  and  closely 
related  compounds  are  to  be  regarded  as  aggregates  of  equal 
numbers  of  positively  and  negatively  charged  atoms. 

Heat  Capacity  of  Solids.  Recent  investigations  of  specific 
heats  of  solids  at  extremely  low  temperatures  have  resulted  in 
the  formulation  of  several  interesting  relationships  between  heat 
capacity  and  temperature. 

At  ordinary  temperatures  the  molecules  of  a  crystalline  solid 
may  be  assumed  to  be  in  a  state  of  violent,  unordered  motion.  As 
the  temperature  is  lowered,  the  amplitude  of  the  molecular  oscil- 
lations steadily  diminishes  until  finally,  at  the  absolute  zero,  there 
is  in  all  probability  a  complete  cessation  of  motion.  In  the  neigh- 
borhood of  the  absolute  zero,  where  the  amplitude  of  the  molecular 
oscillations  is  negligible,  a  crystalline  solid  may  be  assumed  to 
possess  the  properties  characteristic  of  a  perfectly  elastic  body. 
In  other  words,  the  crystalline  forces  holding  the  molecules  to- 
gether would  preponderate  over  the  feeble  thermal  forces  tending 
to  initiate  molecular  oscillations  within  the  solid.  Under  these 
conditions  the  solid  as  a  whole  would  exhibit  the  same  behavior  as 
a  single  molecule,  that  is  to  say,  the  solid  would  function  as  a 
perfectly  elastic  body. 


96  THEORETICAL  CHEMISTRY 

On  this  assumption  Debye*  has  derived  the  following  equation 
expressing  the  heat  capacity  of  a  solid,  Cv,  in  terms  of  its  absolute 
temperature  T, 

>T3 

(i«71.9~-  (1) 

In  this  equation  6  is  a  constant  characteristic  of  each  solid  and  has 
the  same  dimensions  as  T.  The  value  of  6  varies  between  the 
limits  6  =  50  for  calcium  and  6  =  1840  for  carbon.  The  agree- 
ment between  the  observed  and  calculated  values  of  Cv  has  been 
found  to  be  excellent  up  to  T  =  6/12. 

When  this  latter  temperature  is  exceeded,  the  molecules  of  the 
solid  begin  to  absorb  more  and  more  heat  energy  and  to  vibrate 
independently  about  their  centers  of  oscillation.  The  failure  of 
Debye's  equation  is  to  be  expected  under  these  conditions  since 
the  solid  is  no  longer  behaving  as  one  large  molecule.  Obviously 
the  lighter  the  molecules  and  the  greater  the  crystalline  forces 
within  the  solid,  the  higher  must  the  temperature  become  before 
the  individual  molecules  can  acquire  appreciable  kinetic  energy. 
This  is  apparent  from  the  familiar  dynamical  principle,  that  the 
kinetic  energy  of  a  vibrating  particle  is  proportional  to  its  mass 
and  to  the  square  of  its  vibration  frequency.  In  the  case  of  lead, 
which  is  a  soft,  malleable  solid  with  a  relatively  low  melting-point, 
it  is  reasonable  to  infer  that  the  crystalline  forces  are  feeble,  and 
consequently  we  should  expect  that  molecular  and  atomic  vibra- 
tions would  be  set  up  at  quite  low  temperatures.  Furthermore, 
since  the  atoms  of  lead  are  extremely  heavy,  their  kinetic  energy 
must  be  great.  The  correctness  of  these  conclusions  is  confirmed 
by  the  fact  that  the  Debye  equation  when  applied  to  lead  has  been 
found  to  hold  only  over  a  very  short  range  of  temperature.  On 
the  other  hand,  the  equation  has  been  found  to  hold  for  the 
diamond  up  to  a  temperature  of  about  200°  absolute.  In  this 
case  we  have  a  solid  in  which  the  crystalline  forces  are  extremely 
powerful  and  in  which  the  atoms  are  relatively  light.  A  fairly 
high  temperature  must  be  attained  before  the  energy  absorbed 
by  the  individual  atoms  of  the  diamond  acquires  appreciable 
magnitude. 

The  absorption  of  energy  by  the  vibrating  molecules  continues 
to  increase  as  the  temperature  is  raised  until  ultimately,  at  the 

*  Ann.  Physik.,  39,  789  (1912). 


SOLIDS  97 

melting-point  of  the  solid,  the  crystalline  forces  become  negligible. 
As  this  temperature  is  approached  therefore,  the  intermolecular 
restraint  becomes  less  and  less  and  the  mean  kinetic  energy  of 
the  molecules  approaches  that  of  the  molecules  of  the  molten  solid. 
As  has  already  been  stated  in  Chapter  I  (p.  10),  Dulong  and 
Petit,  in  1819,  discovered  the  interesting  fact  that  the  atomic  heats 
of  the  solid  elements  have  a  constant  value  of  6.5.  The  importance 
of  this  generalization  in  connection  with  the  verification  of  atomic 
weights  has  already  been  pointed  out.  Lewis*  has  directed  at- 
tention to  the  fact  that  it  is  much  more  rational  to  calculate 
the  atomic  heat  of  an  element  from  the  specific  heat  at  con- 
stant volume  rather  than  from  the  specific  heat  at  constant 
pressure.  While  it  is  impossible  to  measure  the  specific  heat  at 
constant  volume,  its  value  may  be  derived  from  the  specific  heat 
at  constant  pressure  by  an  application  of  the  laws  of  thermo- 
dynamics. Thus,  Lewis  has  obtained  the  formula 

Ta2V 
C"~  C'=4L78T 

where  T  denotes  the  absolute  temperature,  a  the  coefficient  of 
expansion,  j3  the  coefficient  of  compressibility,  Cp  and  Cv  the  atomic 
specific  heats  at  constant  pressure  and  constant  volume  respec- 
tively and  V  the  atomic  volume.  By  means  of  this  equation, 
Lewis  has  established  the  following  generalization :  Within  the 
limits  of  experimental  error,  the  atomic  heat  at  constant  volume, 
at  20°  C.,  is  the  same  for  all  solid  elements  whose  atomic  weights  are 
greater  than  that  of  potassium,  and  is  equal  to  5.9.  In  the  case  of  a 
solid  having  a  high  melting-point,  the  violent  agitation  of  its  con- 
stituent molecules  and  atoms  as  the  temperature  is  raised,  will 
undoubtedly  produce  a  corresponding  increase  in  the  amplitude  of 
vibration  of  its  electrons  together  with  an  increase  in  their  trans- 
lational  velocity  among  the  molecules.  Under  these  conditions, 
the  specific  heat  at  constant  volume  should  be  greater  than  5.9 
calories.  This  conclusion  cannot  be  verified  experimentally  until 
the  values  of  a  and  /?  in  the  foregoing  equation  have  been  deter- 
mined at  high  temperatures. 

The  complete  heat  capacity  curves  for  three  typical  solid  ele- 
ments, lead,  aluminium,  and  carbon,  are  given  in  Fig.  32.     It  is 
apparent  from  these  curves  that  the  absorption  of  heat  energy  by 
*  Jour.  Am.  Chem.  Soc.,  29,  1165  (1907). 


98  THEORETICAL  CHEMISTRY 

a  crystalline  solid  may  be  considered  as  taking  place  in  three  dis- 
tinct stages,  as  follows: —  (1)  In  the  neighborhood  of  the  abso- 
lute zero,  the  heat  capacity  remains  practically  zero;  (2)  the  heat 
capacity  increases  rapidly  with  the  temperature;  and  (3)  the  heat 
capacity  increases  slowly,  approaching  asymptotically  the  limiting 
value  5.9  for  T7  =  oo .  For  a  malleable,  low  melting  element  of 
high  atomic  weight,  such  as  lead,  the  first  two  stages  are  very 


500 


short  and  the  final  stage  commences  at  a  low  temperature.  On 
the  contrary,  with  a  hard,  high  melting  element  of  low  atomic 
weight,  such  as  carbon  in  the  form  of  diamond,  the  final  stage  is 
not  reached  at  any  temperature  within  the  range  covered  by  the 
experiments. 

The  Nernst-Lindemann  Equation.  Recently,  several  equa- 
tions have  been  derived  expressing  the  heat  capacity  of  a  solid  in 
terms  of  temperature  and  arbitrary  constants.  All  of  these 
equations  are  based  upon  the  so-called  "  quantum  theory  "  accord- 
ing to  which  the  absorption  of  heat  energy  by  matter  is  supposed 
to  take  place  in  a  discontinuous  manner,  the  discrete  units  of 
energy  being  termed  quanta.  While  the  discussion  of  the  quantum 
theory,  and  the  equations  connecting  heat  capacity  and  tempera- 
ture, lie  outside  of  the  scope  of  this  book,  mention  should  never- 
theless be  made  of  the  empirical  equation  derived  by  Nernst  and 
Lindemann.*  This  equation  gives  values  of  Cv  which  are  in  re- 

*  Zeit.  Elektrochem.,  17,  817  (1911). 


SOLIDS  99 


markably  close  agreement  with  the  values  determined  by  direct 
experiment.    The  equation  may  be  written  in  the  following  form : 

(jy •*  (A 


In  this  expression,  R  is  the  molecular  gas  constant  R  =  2  calories, 
e  is  the  base  of  the  natural  system  of  logarithms  and  6  is  a  con- 
stant depending  upon  the  nature  of  the  solid. 

The  value  of  6  may  be  calculated  with  a  fair  degree  of  accuracy 
by  means  of  the  equation 


•  -  •-      - 

in  which  Tf  denotes  the  absolute  melting-point  of  the  substance, 
A  its  atomic  weight  and  d  its  density. 

Liquid  Crystals.  In  addition  to  possessing  well-defined  geo- 
metrical forms,  crystalline  substances  are  characterized  by  their 
resistance  to  deformation  when  subjected  to  mechanical  stress, 
and  by  the  property  of  melting  sharply  at  definite  temperatures 
with  the  production  of  transparent  liquids. 

In  1888,  two  substances,  cholesteryl  acetate  and  cholesteryl 
benzoate,  were  found  by  Reinitzer*  to  behave  in  an  anomalous 
manner  when  heated.  At  definite  temperatures  these  substances 
melted  to  turbid  liquids  which,  in  turn,  became  clear  on  further 
heating,  the  latter  change  also  taking  place  at  definite  tempera- 
tures. On  cooling  the  clear  liquids,  the  reverse  series  of  changes 
was  found  to  occur. 

Examination  of  the  turbid  liquids  revealed  the  fact  that  they 
resembled  ordinary  liquids  in  their  general  behavior,  such  as  assum- 
ing the  spherical  shape  when  suspended  in  a  medium  of  the  same 
density,  or  of  rising  in  a  capillary  tube  under  the  influence  of  sur- 
face tension.  But  in  addition  to  possessing  the  properties  char- 
acteristic of  the  liquid  state,  Lehmann  discovered  that  they  pos- 
sessed optical  properties  which  had  hitherto  been  observed  only 
with  solid,  crystalline  substances.  Their  behavior  towards  polar- 
ized light  was  such  as  to  warrant  tjie  conclusion  that  these  turbid 
liquids  are  anisotropic.  In  view  of  these  facts,  Lehmann  pro- 

*  Monatshefte,  9,  435  (1888). 


100  THEORETICAL  CHEMISTRY 

posed  that  liquids  possessing  these  properties  should  be  called 
liquid  crystals,  the  term  implying  that  under  ordinary  condi- 
tions, the  crystalline  forces  in  these  substances  are  so  feeble  that 
the  crystals  readily  undergo  deformation  and  actually  flow  like 
liquids.  That  these  turbid  liquids  are  not  emulsions,  is  proven 
by  the  fact  that  when  they  are  examined  under  the  microscope, 
the  turbidity  is  found  to  be  due  to  the  aggregation  of  a  myriad 
of  differently  oriented  transparent  crystals.  All  subsequent 
investigation  of  liquid  crystals  has  failed  to  show  any  lack  of 
homogeneity.  The  number  of  such  substances  known  at  the 
present  time  is  fairly  large. 

REFERENCES 
X-Rays  and  Crystal  Structure,  W.  H.  Bragg  and  W.  L.  Bragg. 


CHAPTER  V 

THE  RELATION   BETWEEN   PHYSICAL  PROPERTIES 
AND   MOLECULAR   CONSTITUTION 

A  large  number  of  interesting  and  important  relationships 
have  been  established  between  the  physical  properties  of  different 
chemical  compounds  and  their  molecular  constitution. 

While  it  is  manifestly  impossible  to  discuss  more  than  a  few  of 
these  relationships  within  the  scope  of  this  book,  it  is  desirable 
that  the  student  should  at  least  be  made  familiar  with  the  general 
character  of  investigations  in  this  important  branch  of  physical 
chemistry. 

Before  proceeding  to  a  brief  consideration  of  several  typical 
examples  illustrative  of  the  dependence  of  physical  properties 
upon  molecular  constitution,  it  should  be  pointed  out  that  there 
are  two  different  ways  in  which  such  relationships  may  be  ap- 
proached* Thus,  we  may  seek  to  determine  the  influence  which 
the  number  and  kind  of  the  atoms  present  in  different  molecules 
exert  upon  some  particular  physical  property;  or,  on  the  other 
hand,  we  may  strive  to  ascertain  how  the  property  in  question 
is  influenced  by  the  manner  in  which  the  atoms  are  arranged  in 
the  molecule.  The  first  method  serves  to  bring  out  the  addi- 
tive character  of  the  property,  whilst  the  second  method  empha- 
sizes its  constitutive  character.  Almost  all  physical  properties 
exhibit  both  additive  and  constitutive  characteristics.  In  ad- 
dition to  additive  and  constitutive  properties,  Ostwald  has  in- 
troduced a  third  type  which  he  calls  colligative  properties;  the 
numerical  value  of  colligative  properties  is  entirely  dependent 
upon  the  number  of  molecules  present.  For  example,  the  vol- 
ume of  a  gas  is  a  colligative  property  since  it  is  practically 
proportional  to  the  total  number  of  molecules  present  in  the  gas. 
Obviously,  since  colligative  properties  are  independent  of  both 
composition  and  structure,  they  call  for  no  further  consideration 
in  this  chapter. 

Molecular  Volume.  In  dealing  with  the  volume  relations  of 
liquids  it  is  customary  to  employ  the  molecular  volume  ie.,  the 

101 


.102  l    .  ;  :         THEORETICAL  CHEMISTRY 

volume  occupied  by  the  molecular  weight  of  the  liquid  in  grams. 
•The  justification  for  this  procedure  is,  that  when  we  compare  the 
gram-molecular  weights  of  liquids,  the  comparison  involves  equal 
numbers  of  molecules  of  the  different  substances. 

Relations  between  the  molecular  volumes  of  liquids  were  first 
pointed  out  by  Kopp.*  On  comparing  the  molecular  volumes  of- 
different  liquids  at  their  boiling-points,  he  found  that  constant 
differences  in  composition  correspond  to  approximately  constant 
differences  in  the  molecular  volumes.  Thus,  the  molecular  vol- 
umes of  the  successive  members  of  an  homologous  series  differ 
by  the  same  number  of  units,  this  difference  corresponding,  for 
example,  to  a  CH2  group.  In  like  manner,  the  molecular  volumes  of 
various  groups  have  been  determined,  and  from  these  in  turn  the 
atomic  volumes  of  the  constituent  elements  have  been  worked  out. 

While  Kopp  found  that  his  results  were  most  regular  when  the 
molecular  volumes  were  determined  at  the  boiling  temperatures 
of  the  respective  liquids,  the  reason  for  this  did  not  appear  until 
after  van  der  Waals  had  developed  his  theory  of  corresponding 
states.  As  has  already  been  pointed  out,  the  boiling-points  of 
most  liquids  are  approximately  two-thirds  of  their  respective 
critical  temperatures,  and  therefore  are  corresponding  tempera- 
tures. 

Subsequent    investigations    have    shown    that    the    numerical 
values  of  the  atomic  volumes  of  the  elements,  as  originally  derived 
by  Kopp,  are  more  or  less  erroneous.     The  following  values  of 
the  atomic  volumes  are  those  recently  published  by  Le  Basf  and 
represent  the  average  values  derived  from  the  study  of  a  large 
number  of  organic  compounds  of  widely  different  character : 
C     =  14.8         Cl  =  24.6         I  =  37.0       N  =  15.6 
H     =    3.7         Br  =  27.0         S  =  25.6  =  10.5  (in  primary 

amines) 

O"  =    7.4  (as  in  aldehydes,  ketones  =  12.0  (in    second- 

and  doubly  linked  oxygen  ary  amines) 

generally) 
O'    =    9.1  (in  methyl  esters) 

=  11.0  (in  higher  esters)     ^0  =  9.9  (in  methyl  ethers) 
=  12.0  (in  acids)  =11.0  (in  higher  ethers) 

*  Lieb.  Ann.,  41,  79  (1842);  96,  153,  303  (1855);  96,  171  (1855). 
t  The  Molecular  Volumes  of  Liquid  Chemical  Compounds, 


RELATION  BETWEEN  PHYSICAL  PROPERTIES 


103 


As  will  be  seen,  the  value  of  the  atomic  volume  is  dependent 
upon  the  manner  of  linkage  of  the  atoms  in  the  molecule. 

By  means  of  such  a  table  of  experimentally  determined  atomic 
volumes  it  is  possible  to  calculate  the  molecular  volumes  of  the 
lower  members  of  the  various  homologous  series  of  organic  liquids 
with  a  fair  degree  of  accuracy.  For  example,  the  molecular 
volume  of  acetic  acid,  C2H4O2,  may  be  calculated  from  the  atomic 
volumes  of  its  constituent  atoms  as  follows :  — 

2C  =2X14.8  =  29.6 
4H  =  4  X  3.7  =  14.8 
1O'  =  1  X  12.0  =  12.0 
1O"  =  1  X  7.4  =  7.4 

Molecular  volume      63 . 8 

The  density  of  acetic  acid  at  its  boiling-point  is  0.942  and  its 
molecular  weight  is  60,  therefore  the  observed  value  of  the  mo- 
lecular volume  is,  60/0.942  =  63.7,  a  result  which  is  in  excellent 
agreement  with  that  calculated  from  the  atomic  volumes  of  the 
constituent  atoms.  The  following  table  gives  the  calculated  and 
observed  values  of  the  molecular  volumes  of  a  number  of  different 
organic  compounds  at  their  boiling-points :  — 

MOLECULAR  VOLUMES 

(Cubic  centimeters  per  mol) 


Substance 

Formula 

Vm  (calc.) 

Vm  (obs.) 

IMethyl  alcohol 

CH4O 

38.7 

42.0 

Ethyl  alcohol 

C2H6O 

60.9 

62.0 

Amyl  alcohol 

C6H12O 

127.5 

124.0 

Acetaldehyde 

C2H4O 

51.8 

56.5 

Acetone                                       

C3H6O 

74.0 

77.5 

Formic  acid                             

CH2O2 

41.6 

41.3 

Acetic  acid                          

C2H4O2 

63.8 

63.7 

Propionic  acid    

C3H6O2 

86.0 

85.4 

Methyl  formate  

C2H4O2 

63.8 

63.4 

Methyl  acetate     

C3H6O2 

86.0 

85.8 

Ethyl  ether  

C4H10O 

106.1 

106.0 

Benzene 

111.0 

96.0 

Naphthalene 

CioHs 

177.6 

149.2 

It  is  apparent  from  the  data  of  the  foregoing  table  that  while  the 
agreement  between  the  observed  and  calculated  values  of  the 


104  THEORETICAL  CHEMISTRY 

molecular  volumes  is  in  some  cases  very  satisfactory,  in  other 
cases  the  differences  are  so  great  as  to  engender  doubt  as  to  the 
purely  additive  character  of  this  property.  In  fact,  recent  re- 
search has  tended  to  restrict  the  application  of  Kopp's  additive 
generalization  and  to  lay  more  emphasis  upon  the  constitutive 
character  of  this  property.  In  this  connection,  Le  Bas  writes:  — 
"  A  study  of  molecular  volumes  has  shown  us  that  probably  we 
are  not  at  the  end  of  the  utility  of  physical  properties  as  a  means 
of  giving  us  an  insight  into  the  structure  of  molecules.  When 
a  more  scientific  method  of  examination  of  these  physical  prop- 
erties shall  have  been  worked  out,  there  is  no  doubt  that  a  great 
advance  will  be  made  in  our  knowledge  of  the  intricacies  of  chem- 
ical constitution.'7 

Co-volume.  By  studying  various  series  of  hydrocarbons, 
alcohols  and  ethers,  Traube*  has  been  led  to  suggest  that  the 
molecular  volume  of  a  liquid  be  looked  upon  as  made  up  of  the 
atomic  volumes  of  its  constituent  elements  together  with  a  quan- 
tity which  he  terms  the  co-volume.  This  latter,  he  defines  as  the 
space  surrounding  a  molecule  within  which  it  is  free  to  vibrate  and 
from  which  other  molecules  are  excluded.  The  co-volume  ap- 
pears to  be  nearly  constant  for  a  large  number  of  substances,  its 
mean  value  at  a  temperature  of  15°  C.  being  25.9  cc.  The  values 
assigned  by  Traube  to  the  atomic  volumes  of  some  of  the  elements 
are  as  follows :  — 

C  =  9.9       0  =    5.5       Br  =  17.7     N  (trivalent)        =     1.5 
H  =  3.1      Cl  =  13.2          I  =  21.4     N  (pentavalent)  =  10.7 

Traube  has  worked  out  a  series  of  constants  which  must  be  de- 
ducted to  allow  for  ring  formation  and  for  double  and  treble  link- 
ing. By  means  of  these  values,  it  is  possible  to  calculate  the 
molecular  volume  of  a  substance  by  adding  together  the  respective 
atomic  volumes  of  the  constituents  of  the  liquid  and  the  co- 
volume,  25.9.  It  is  of  course  necessary  to  know  the  molecular 
weight  of  the  substance  together  with  its  constitution,  so  that 
due  allowance  may  be  made  for  unsaturation.  For  example, 

*  Uber  den  Raum  der  Atome.  J.  Traube.  Ahrens'  Sammlung  Chemischer 
und  chemisch-technischer  Vortraege,  4,  255  (1899). 


RELATION  BETWEEN  PHYSICAL  PROPERTIES  105 

the  molecular  volume  of  ethyl  ether,  C4Hi0O,  may  be  calculated 
by  Traube's  method  as  follows :  — 

4C  =    4  X9.9  =  39.6 
10  H  =  10  X3.1  =  31 
1O=     1  X  5.5  =    5.5 

76.1 
Co-volume  25.9 


Molecular  volume  102.0 

The  molecular  volume,  as  determined  from  the  molecular  weight 
and  density  at  15°  C.,  is  74  -j-  0.7201  =  102.7. 

The  method  of  Traube  may  be  employed  in  roughly  checking 
the  accepted  value  of  the  molecular  weight  of  a  liquid,  provided 
its  density  at  15°  C.  is  known,  since  in  the  equation 

M/d  =  S  atomic  volumes  +  25.9,  (1) 

M  is  the  only  unknown  quantity.  It  is  apparent  that  the  liquid 
must  be  non-associated,  since  for  an  associated  substance  the 
normal  co-volume  must  necessarily  accompany  the  polymerized 
molecule.  In  this  case  the  formula  becomes 

M/d  =  S  atomic  volumes  +  25.9/n, 

where  n  denotes  the  number  of  simple  molecules  in  the  polymer. 
Obviously  when  the  molecular  weight  of  a  liquid  is  known,  the 
experimental  determination  of  the  co-volume,  (M/d  —  2  atomic 
vols.)  may  be  used  to  estimate  the  degree  of  association.  The 
values  thus  obtained  are  not  in  satisfactory  agreement  with  the 
factors  of  association  derived  by  means  of  other  methods. 

Refractive  Power  of  Liquids.  The  velocity  of  transmission 
of  light  through  any  medium  depends  upon  its  nature,  especially 
upon  its  density.  When  a  ray  of  light  passes  from  one  medium 
into  another  it  is  refracted,  the  degree  of  refraction  being  such 
that  the  ratio  of  the  sines  of  the  angles  of  incidence  and  refrac- 
tion is  constant  and  characteristic  for  the  two  media.  This 
fundamental  law  of  refraction  was  discovered  by  Snell  about 
1621.  According  to  the  wave  theory  of  light,  the  ratio  of  the 
sines  of  the  angles  of  incidence  and  refraction  is  identical  with 
the  ratio  of  the  velocities  of  light  in  the  two  media.  The  ratio  is 
termed  the  index  of  refraction  and  is  usually  denoted  by  the  letter 


106 


THEORETICAL  CHEMISTRY 


n.  Representing  by  i  and  r,  the  angles  of  incidence  and  refraction, 
and  by  vi  and  vz,  the  respective  velocities  of  light  in  the  two  media, 
we  have 

sin  i      v\  /0x 

(Z) 


n 


sn  r 


Various  forms  of  apparatus  have  been  devised  for  the  determina- 
tion of  the  refractive  index  of  liquids.     Of  these  the  best  known 


Fig.  33 

and  most  satisfactory  is  the  refractometer  of  Pulfrich,  an  improved 
form  of  which  is  shown  in  Fig.  33.  While  the  limits  of  this  book 
prohibit  a  detailed  description  of  the  apparatus,  the  fundamental 
principles  involved  in  its  construction  will  be  readily  understood 


RELATION  BETWEEN  PHYSICAL  PROPERTIES  107 


from  the  accompanying  diagram,  Fig.  34.  The  liquid  or  fused 
solid  is  placed  in  a  small  glass  cell,  C,  which  is  cemented  to  a  rec- 
tangular prism  of  dense  optical  glass,  P,  the  refractive  index  of 
which  is  generally  1.61.  A  beam  of  monochromatic  light,  from 
a  sodium  flame,  or  a  spectrum-tube  containing  hydrogen,  is  al- 
lowed to  enter  the  prism  in  a  direction  parallel  to  the  horizontal 
surface  of  separation  between 
the  glass  and  the  liquid.  After 
passing  through  the  liquid  and 
the  prism,  the  beam  emerges 
making  an  angle  i  with  its  orig- 
inal direction.  By  means  of  a 
telescope,  the  emergent  beam 
can  be  observed  and  its  position 
noted,  the  angle  of  emergence 
being  read  on  a  divided  circle 


Fig.  34 


attached  to  the  telescope.  From  the  angle  of  emergence  thus 
determined,  the  index  of  refraction  of  the  liquid  can  be  calculated 
in  the  following  manner.  The  value  of  the  index  of  refraction,  N, 
for  air/glass  being  known,  we  have 


•,r 

N  = 


sin  i 
sin  r 


(3) 


The  angle  of  incidence  of  the  last  ray  entering  the  prism  from  the 
liquid  is  90°,  or  sin  i±  =  1.  The  index  of  refraction,  HI,  for  liquid/ 
glass  may  be  calculated  thus, 

sin  i\         1 


But 


sin 


sin  7*1 
Vl  —  sin2  r. 


(4) 


sin  rt  =  cos  r  =  v  I  —  sin-'  r.  (5) 

Transposing  equation  (3)  and  substituting  in  equation  (5),  we 
have 


sin 


sn 


or 


sn 


=  jr  V  N2  -  sin2  i. 


Therefore,  substituting  equation  (6)  in  equation  (4),  we  have 

N 


(6) 


(7) 


108  THEORETICAL  CHEMISTRY 

Remembering  that  n  =  N/n\,  we  have  for  the  index  of  refraction, 
n,  for  air/liquid,  by  substitution  in  equation  (7), 


n  =  VN2  -  sin2*,  (8) 

or  if  N  =  1.61, 

n  =  V2.5921  -  sin2  i.  (9) 

The  values  of  V  N2  —  sin2  i  are  generally  given,  for  different 
values  of  i,  in  tables  supplied  with  the  refractometer,  thus  saving 
the  experimenter  a  somewhat  laborious  calculation.  The  value 
of  n  thus  obtained  is  the  index  of  refraction  from  air  into  the 
liquid;  if  the  index  from  vacuum  into  the  liquid,  the  so-called 
absolute  index,  is  required,  the  value  of  n  must  be  multiplied  by 
1.00029. 

The  index  of  refraction  is  dependent  upon  temperature,  pres- 
sure and  in  general  upon  all  conditions  which  affect  the  den- 
sity of  the  medium.  Furthermore,  it  is  dependent  upon  the 
wave-length  of  the  light  employed,  the  index  for  the  red  rays 
being  less  than  that  for  the  violet  rays.  It  is  therefore  necessary 
in  making  measurements  of  refractive  indices  to  use  light  of  a 
definite  wave-length,  or  what  is  termed  monochromatic  light.  The 
sodium  flame  is  most  frequently  used  for  this  purpose,  the  wave- 
length being  represented  by  the  letter  D.  Measurements  of  the 
refractive  index  referred  to  the  D-line  of  sodium  are  commonly 
designated  by  the  symbol  nD.  When  incandescent  hydrogen  is 
employed  as  a  source  of  light,  the  refractive  index  may  be  deter- 
mined for  the  C-,  F-  and  G-lines,  the  respective  values  being 
represented  by  nc,  nF,  and  nG. 

Specific  and  Molecular  Refraction.  Various  attempts  have 
been  made  to  express  the  refractive  power  of  a  liquid  by  a  formula 
which  is  independent  of  variations  of  temperature  and  pressure. 
Of  the  different  formulas  proposed  but  two  need  be  mentioned. 
The  first,  due  to  Gladstone  and  Dale,*  is  as  follows:  — 


in  which  d  denotes  the  density  of  the  liquid  and  r\  is  the  so-called 
specific  refraction.     The  other  formula,  proposed  by  Lorenz  f  and 


*  Phil.  Trans.  (1858). 

t  Wied.  Ann.,  n,  70  (1880). 


RELATION  BETWEEN  PHYSICAL  PROPERTIES          109 

Lorentz,*  has  the  following  form:  —  / 

_  1     n*  -  1 

~5'^T2* 

This  formula  is  superior  to  that  of  Gladstone  and  Dale  which  is 
purely  empirical.  It  is  based  upon  the  electromagnetic  theory 
of  light  and  gives  values  of  r2  which  are  quite  independent  of  the 
temperature.  In  order  that  we  may  compare  the  refractive 
powers  of  different  liquids,  the  specific  refractions  are  multiplied 
by  their  respective  molecular  weights,  the  resulting  products 
being  termed  their  molecular  refractions.  As  the  result  of  a  large 
number  of  experiments,  it  has  been  shown  that  the  molecular 
refraction  of  a  compound  is  made  up  of  the  sum  of  the  refractive 
constants  of  the  constituent  atoms,  or  in  other  words  refractive 
power  is  an  additive  property.  The  values  of  the  refractive  con- 
stants  of  the  elements  and  commonly-occurring  groups  have  been 
determined  with  great  care  by  Briihl  and  others,  the  method 
employed  being  similar  to  that  used  by  Kopp  in  connection  with 
his  investigations  on  molecular  volumes.  Thus,  Briihl  found  in 
the  homologous  series  of  aliphatic  compounds  that  a  difference 
of  CH2  in  composition  corresponds  to  a  constant  difference  of 
4.57  in  molecular  refraction.  Then,  having  determined  the  mo- 
lecular refraction  of  a  ketone  or  an  aldehyde  of  the  composition, 
CnH2nO,  he  subtracted  n  times  the  value  of  CH2  and  obtained  the 
atomic  refraction  of  carbonyl  oxygen.  By  deducting  the  molecular 
refraction  of  the  hydrocarbon,  CnH2n+2,  from  that  of  the  corre- 
sponding alcohol,  CnH2n+2O,  he  obtained  the  atomic  refraction  of 
hydroxyl  oxygen.  By  subtracting  six  times  the  value  of  CH2 
from  the  molecular  refraction  of  hexane,  CeHi4,  he  obtained  the 
refractive  constant  for  hydrogen  or  2  H  =  2.08.  In  like  manner 
the  refractions  of  other  elements  and  groups  of  elements  were 
determined. 

Just  as  in  the  case  of  molecular  volumes  so  with  molecular 
refractions,  the  arrangement  of  the  atoms  in  the  molecule  must 
be  taken  into  consideration.     Bruhl,f  who  devoted  much  time  to 
the  investigation  of  the  effect  of  constitution  upon  refraction, 
pointed  out  that  the  molecular  refraction  of  compounds  contain- 
ing double  and  triple  bonds  is  greater  than  the  calculated  value, 
*  Ibid.,  9,  641  (1880). 
t  Troc.  Roy.  Inst.,  18,  122  (1906). 


110 


THEORETICAL  CHEMISTRY 


and  he  assigned  to  these  bonds  definite  constants  of  refraction. 
The  values  of  the  atomic  refractions  for  a  few  of  the  elements 
are  given  in  the  following  table.* 


ATOMIC  REFRACTIONS 


Element 

Gladstone  and  Dale 
Formula 
RD 

Lorenz  and  Lorentz 
Formula 
R'D 

Carbon  singly  bound  and  occurring 
alone  C* 

4.71 

2.592 

Carbon  singly  bound,  C'   .       

4.71 

2.051 

Hydrogen,  H                 

1.47 

1.051 

Hydroxyl  oxygen,  O'  

2.65 

1.521 

Ethereal  oxygen  O^ 

2  65 

1  683 

Ketonic  oxygen  O" 

3  33 

2  287 

Double  bond    = 

2  64 

1  707 

Triple  bond   = 

2  10 

Chlorine  Cl 

10  05 

5  998 

Bromine,  Br            .... 

15.34 

8.927 

Iodine,  I  .    .  .             

25.01 

14.12 

When  applied  to  simple  compounds  of  carbon,  hydrogen  and 
oxygen  the  agreement  between  the  observed  and  calculated 
values  of  molecular  refractions  is  generally  very  satisfactory. 
For  example,  we  may  compare  the  calculated  and  observed  values 
of  the  molecular  refraction  of  lactic  acid,  employing  the  Lorenz- 
Lorentz  formula.  The  formula  of  lactic  acid  is,  CH3CHOHCOOH, 
or  CaHeCVO",  hence  the  molecular  refraction  is  calculated  as 
follows :  — 

3  C  =  3  X  2.501  =    7.503 

6  H  =  6  X  1.051  =    6.306 

2  0'  =  2  X  1.521  =    3.042 

10"  =  1  X  2.287  =    2.287 

Molecular  refraction  =  19.138 

The  observed  value  of  the  molecular  refraction  of  lactic  acid  is 
19.138. 

It  has  been  found  that  the  refractive  power  of  a  substance  is 
influenced  not  only  by  the  presence  of  double  and  triple  bonds,  but 

*  In  this  connection  the  student  should  consult  a  recent  paper  entitled 
"  Atomic  Refraction  "  by  W.  Swietoslawski  in  the  Journal  of  the  American 
Chemical  Society,  42,  1945,  (1920). 


RELATION  BETWEEN  PHYSICAL  PROPERTIES 


111 


also  by  the  relative  position  of  such  bonds  within  the  molecule. 
When  a  molecule  contains  only  one  double  or  triple  bond,  the 
molecular  refraction  may  still  be  calculated  with  considerable 
accuracy  from  the  values  of  the  atomic  refractions,  due  allow- 
ance being  made  for  the  condition  of  unsaturation.  When, 
however,  a  second  double  or  triple  bond  is  introduced  into  the 
molecule  in  a  position  adjacent  to  the  first,  the  calculated  value 
of  the  molecular  refraction  is  found  to  differ  appreciably  from  the 
observed  value.  This  phenomenon  is  known  as  optical  anomaly. 
In  most  cases  the  introduction  of  a  second  unsaturated  group 
increases  the  molecular  refraction  and  the  substance  is  said  to 
exhibit  optical  exaltation;  in  a  few  cases,  however,  the  refractive 
power  is  diminished  and  the  effect  is  then  known  as  optical  de- 
pression. 

Where  two  unsaturated  groups  are  separated  by  one  or  more 
saturated  groups,  optical  anomaly  is  generally  absent.  Where 
unsaturated  groups  occupy  such  positions  within  the  molecule 
that  they  exert  mutual  influence,  they  are  said  to  be  "  conjugated." 
The  anomalous  behavior  of  organic  compounds  containing  con- 
jugated groups  has  been  attributed  by  Thiele  to  the  residual 
affinity  or  partial  valence  of  the  unsaturated  carbon  atoms. 
The  following  table  gives  the  observed  and  calculated  values  of 
the  molecular  refraction  of  several  organic  compounds  which 
exhibit  optical  exaltation. 

MOLECULAR  REFRACTIONS  OF  SOME 
UNSATURATED  COMPOUNDS 


Substance 

Formula 

MR'D 

(calc.) 

MR'D 

(obs.) 

Diff. 

Mesityl  oxide 

(CH3)2C  =  CH-C-CHa 

29.51 

30  37 

0  86 

Benzalacetone  

O 
C6H6-CH  =  CH-C-CH3 

44.64 

48.61 

3.97 

Benzalmethylethyl      ke- 
tone 

ft 

C6H5-CH  =  CH-C-CH2-CH3 

49.24 

53  08 

3  84 

Benzalpinacolin  

A 

C6H6-CH  =  CH-C-C(CH3)3 

58.44 

62.77 

4.33 

ft 

112  THEORETICAL  CHEMISTRY 

The  determination  of  the  molecular  refraction  of  a  liquid 
affords  a  means  of  ascertaining  or  confirming  its  chemical  con- 
stitution. For  example,  geraniol  has  the  formula  doHigO,  and 
its  chemical  behavior  is  such  as  to  warrant  the  conclusion  that  it 
is  a  primary  alcohol.  The  value  of  UD  is  1.4745,  from  which  the 
molecular  refraction  is  calculated  to  be  48.71.  The  molecular 
refraction  calculated  from  the  atomic  refractions  given  in  the 
preceding  table  is  :  — 

10  C  =  10  X  2.501  =  25.01 

18  H  =  18  X  1.051  =  18.92 

1  Hydroxyl  O  =     1  X  1.521  =     1.52 

Molecular  refraction       45.45 

The  difference  between  the  theoretical  and  experimental  values 
of  the  molecular  refraction  is  48.71  —  45.45  =  3.26,  which  is 
approximately  twice  the  value  of  a  double  bond,  1.707  X  2  = 
3.414.  From  this  we  conclude  that  the  molecule  of  geraniol  con- 
tains two  double  bonds.  Furthermore  an  alcohol  of  the  formula, 
CioHigO,  containing  two  double  bonds  cannot  possess  a  ring 
structure  and  therefore  must  be  a  member  of  the  aliphatic  group 
of  compounds.  This  conclusion  is  supported  by  the  chemical 
properties  of  the  substance.  The  accepted  structural  formula 
of  geraniol  is 

H  -  C  -  CH2OH 

(CH3)2C  =  CH-CH2-CH2  -  C  -  CH3. 

In  a  similar  manner  the  Kekule  formula  for  benzene  has  been 
confirmed,  the  difference  between  the  theoretical  and  experimental 
values  of  the  molecular  refraction  indicating  the  presence  of  three 
double  bonds  in  the  molecule. 

Thus,  Briihl*  found  that  on  changing  from  an  acetylene  to  an 
ethylene  linkage  there  is  a  decrease  of  0.40  units  in  molecular 
refractive  power,  and  also  that  in  the  polymerization  of  acetylene 
to  benzene,  as  represented  by  the  equation, 

3  C2H2 


a  decrease  of  1.19  units  in  refractive  power  occurs.  Since  the 
decrease  in  refractive  power  is  approximately  three  times  the 
change  in  refractive  power  resulting  from  the  replacement  of  a 

*  Zeit.  phys.  Chem.,  i,  343  (1887). 


RELATION  BETWEEN  PHYSICAL  PROPERTIES          113 

triple  by  a  double  bond,  Briihl  inferred  that  in  a  molecule  of 
benzene  there  must  be  three  double  bonds  as  in  the  familiar  for- 
mula proposed  by  Kekule. 

Specific  Refraction  of  Mixtures.  The  specific  refraction  of  an 
homogeneous  mixture  or  solution  is  the  mean  of  the  specific 
refractions  of  its  constituents.  Thus,  if  the  specific  refractions 
of  the  mixture  and  its  two  components  are  represented  by  r\t  r2, 
and  r3,  then 


where  p  denotes  the  percentage  of  the  constituent  whose  specific 
refraction  is  r2.  Hence  it  is  possible  to  determine  the  specific 
refraction  of  a  substance  in  solution  by  measuring  the  refractive 
indices  and  densities  of  the  solution  and  solvent.  If  the  refractive 
indices  of  the  solvent,  solution  and  dissolved  substance  are  repre- 
sented by  HI,  n-2,  and  n3  respectively,  and  if  di,  d2,  and  ds  denote 
the  corresponding  densities,  then  we  have 

1_  ns2  -  1  =  100   ?i22  -  1  _  100  -p  ni2  -  1 
d3*n32  +  2~d2p'n22  +  2          dlP    "Wl2  +  2' 

where  p  is  the  percentage  of  the  dissolved  substance. 

As  has  already  been  mentioned,  the  formula  of  Lorenz-Lorentz  is 
based  upon  the  electromagnetic  theory  of  light.  According  to  this 
theory  n2  —  1/n2  +  2  expresses  the  fraction  of  the  unit  of  volume 
of  the  substance  which  is  actually  occupied  by  it.  From  this  it 

M  ri2  —  I 
follows  that  the  molecular  refraction,  -=  ---  2         ,  is  an  expression 

•>    d    n   -\-  2 

of  the  volume  actually  occupied  by  the  atomic  nuclei  of  the 
molecule.  It  is  interesting  to  note  that  the  ratio  of  the  sum  of 
the  atomic  volumes,  calculated  by  the  method  of  Traube,  to  the 
corrected  molecular  volume,  as  determined  by  the  Lorenz-Lorentz 
formula,  is  approximately  constant,  or 

2  atomic  volumes      0  .  _ 

-  TT  —  £  -  -  -  =  3.45  approximately.  (14) 


This  may  be  considered  as  the  ratio  of  the  volume  within  which 

the  atoms  execute  their  vibrations  to  their  actual  material  volume. 

Rotation  of  the  Plane  of  Polarized  Light.     Some  liquids  when 

placed  in  the  path  of  a  beam  of  polarized  light  possess  the  prop- 


114  THEORETICAL  CHEMISTRY 

erty  of  rotating  the  plane  of  polarization  to  the  right  or  to  the 
left.  Such  liquids  are  said  to  be  optically  active.  Those  substances 
which  rotate  the  plane  of  polarization  to  the  right  are  termed 
dextro-rotatory,  while  those  which  cause  an  opposite  rotation  are 
called  levo-rotatory .  The  determination  of  the  rotatory  power  of 
a  liquid  is  made  by  means  of  an  instrument  known  as  a  polarimeter, 
a  convenient  form  of  which  is  shown  in  Fig.  35.  The  essential 


Fig.  35 

parts  of  this  instrument  are  two  similar  Nicol  prisms  placed  one 
behind  the  other  with  their  axes  in  the  same  straight  line.  The 
light  after  passing  through  the  forward  prism,  P,  known  as  the 
polarizer,  has  its  vibrations  reduced  to  a  single  plane;  it  is  said 
to  be  plane  polarized.  On  entering  the  rear  Nicol  prism,  A, 
known  as  the  analyzer,  the  light  will  either  pass  through  or  be 
completely  stopped,  depending  upon  the  position  of  the  prism. 
If  the  analyzer  be  slowly  rotated,  it  will  be  observed  that  the 
positions  of  maximum  transmission  and  extinction  occur  at  points 
90°  apart.  If  the  analyzer  be  rotatedfj"  so  that  its  axis  is  at  right 
angles  to  the  axis  of  the  polarizer,  the  field  observed  will  be  dark, 
no  light  being  transmitted.  If  now  a  tube  similar  to  that  shown 
in  Fig.  36  be  filled  with  an  optically  active  liquid  and  placed 
between  the  polarizer  and  analyzer,  the  field  will  become  light 


RELATION  BETWEEN  PHYSICAL  PROPERTIES          115 

again,  due  to  the  rotation  of  the  plane  of  polarization  by  the 
optically-active  substance.  The  extent  to  which  the  plane  of 
polarization  has  been  rotated  can  be  determined  by  turning  the 
analyzer  until  the  field  becomes  dark  again,  and  reading  on  the 
divided  circle,  K,  the  number  of  degrees  through  which  it  has 
been  moved.  When  it  is  necessary  to  turn  the  analyzer  to  the 


Fig.  36 

right,  the  substance  is  dextro-rotatory,  and  when  it  is  necessary 
to  turn  it  to  the  left,  the  substance  is  levo-rotatory.  Various 
optical  accessories  have  been  added  to  the  simple  polarimeter 
described  above  to  render  the  instrument  more  sensitive,  but  for 
these  details  the  student  must  consult  some  special  treatise.* 
The  angle  of  rotation  is  dependent  upon  the  nature  of  the  liquid, 
the  length  of  the  column  of  substance  through  which  the  light 
passes,  the  wave-length  of  the  light  used,  and  the  temperature  at 
which  the  measurement  is  made.  It  is  customary  in  polarimetric 
work  to  employ  sodium  light  and,  unless  otherwise  specified,  it 
may  be  assumed  that  a  given  rotation  corresponds  to  the  D-line. 

Specific  and  Molecular  Rotation.  The  results  of  polarimetric 
measurements  are  expressed  either  as  specific  rotations  or  as 
molecular  rotations,  the  latter  being  preferable  since  the  optical 
activities  of  different  substances  may  then  be  compared. 

The  specific  rotation  is  obtained  by  dividing  the  observed  rota- 
tion by  the  product  of  the  length  of  the  column  of  liquid  and  its 
density,  or 


where  [a]t  is  the  specific  rotation  at  the  temperature  t,  a  the 
observed  angle,  I  the  length  of  the  column  of  liquid  in  decimeters, 
and  d,  its  density.  If  the  specific  rotation  is  multiplied  by  the 
molecular  weight  of  the  substance,  the  molecular  rotation  is  ob- 
tained, but  owing  to  the  fact  that  the  resulting  numbers  are  too 
large,  it  is  customary  to  express  the  molecular  rotation  as  one 

*  See  for  example,  "  The  Optical  Rotatory  Power  of  Organic  Substances 
and  its  Practical  Applications."     H.  Landolt,  trans,  by  J.  H.  Long. 


116  THEORETICAL  CHEMISTRY 

one-hundredth  of  this  value,  thus 


Ma  ,1Ax 

' 


The  specific  and  molecular  rotations  of  solutions  of  optically 
active  substances  may  also  be  determined,  if  we  assume  that  the 
solvent  is  without  effect.  While  this  assumption  is  justifiable 
with  aqueous  solutions,  it  is  not  so  when  non-aqueous  solvents  are 
used.  If  g  grams  of  an  optically  active  substance  be  dissolved  in 
v  cc.  of  solvent,  then 

av  M     av  ,17N 

[«],  =  -and        [«„],.  _., 


or,  if  the  composition  of  the  solution  is  expressed  in  terms  of 
weight  instead  of  volume,  g  grams  of  substance  being  dissolved 
in  100  grams  of  solution  of  density  d,  then 

100  a  Ma  /icx 

a]i  =  ~  laMl<==- 


Optical  Activity  and  Chemical  Constitution.  The  fact  that 
some  substances  have  the  power  of  rotating  the  plane  of  polarized 
light  was  first  discovered  by  Biot,  but  the  credit  for  recognizing 
the  chemical  significance  of  this  fact  belongs  to  Pasteur.*  He 
discovered  that  ordinary  racemic  acid  can  be  separated  into  two 
optically  active  modifications,  one  of  which  is  dextro-  and  the 
other  levo-rotatory,  the  numerical  values  of  the  two  rotations 
being  identical.  If  a  solution  of  sodium  ammonium  racemate 
be  allowed  to  evaporate  at  a  low  temperature,  crystals  of  the 
composition  NaNH4C4H4O6.4  H2O  will  separate.  On  close  in- 
spection it  will  be  found  that  the  crystals  are  not  all  alike,  but 
that  they  may  be  divided  into  two  classes,  one  class  showing  some 
unsymmetrical  crystal  surfaces  which  are  oppositely  placed  ir 
the  crystals  of  the  other  class.  The  crystals  of  one  class  may 
be  regarded  as  the  mirror  images  of  those  of  the  other  class: 
such  crystals  are  said  to  be  enantiomorphous.  The  forms  usually 
assumed  by  the  two  enantiomorphous  modifications  of  sodium 
ammonium  racemate  are  shown  in  Fig.  37.  After  separating 
the  two  forms  Pasteur  dissolved  each  in  water,  making  the  solu- 
tions of  the  same  strength.  The  solution  of  the  crystals  with  the 

*  Ann.  Chim.  Phys.  (3),  24,  442  (1848);  28,  56  (1850);  31,  67  (1851). 


RELATION   BETWEEN   PHYSICAL  PROPERTIES 


117 


"  right-handed  faces  "  was  found  to  be  dextro-rotatory,  while  that 
of  the  crystals  with  the  "  left-handed  faces  "  was  found  to  be 
levo-rotatory.  Pasteur  then  decomposed  the  two  salts  obtained 
from  sodium  ammonium  racemate  and  obtained  the  corresponding 
acids,  which  he  called  dextro-  and  levo-racemic  acids.  It  was 
subsequently  shown  that  the  two  acids  were  identical  with  dextro- 
and  levo-tartaric  acids.  Finally,  when  Pasteur  mixed  equiv- 
alent amounts  of  concentrated  solutions  of  dextro-  and  levo- 
tartaric  acids,  an  appreciable ___^ 

evolution  of  heat  was  ob- 
served, indicating  that  a 
chemical  reaction  had  taken 
place.  After  allowing  the 
solution  to  stand  for  some 
time,  crystals  of  ordinary 
racemic  acid  were  obtained. 


Fig.  37 


Thus  it  was  clearly  proven  that  an  optically  inactive  substance 
may  be  separated  into  two  optically  active  modifications,  possess- 
ing equal  and  opposite  rotatory  powers,  and  that  by  mixing 
equivalent  quantities  of  the  two  optically  active  forms,  the 
optically  inactive  substance  may  be  recovered. 

Pasteur  discovered  and  applied  three  other  methods,  in  addi- 
tion to  the  mechanical  method  already  described,  for  the  separation 
of  a  substance  into  its  optically  active  modifications.  These  are 
as  follows :  —  (a)  Method  of  Crystallization;  (b)  Method  of  Forma- 
tion of  Derivatives;  and  (c)  Methods  of  Ferments. 

Methods  of  Crystallization.  To  a  supersaturated  solution  of  the 
racemic  modification,  a  very  small  crystal  of  one  of  the  active 
forms  is  added.  This  will  induce  the  separation  of  crystals  of 
the  same  form,  inoculation  with  a  dextro-crystal  producing  the 
dextro-form  and  inoculation  with  a  levo-crystal  producing  the 
levo-form. 

Method  of  Formation  of  Derivatives.  In  this  method  an  opti- 
cally active  substance,  generally  an  alkaloid,  is  added  to  the 
racemic  modification,  producing  optically  active  derivatives  having 
different  solubilities.  Thus  if  cinchonine,  an  optically  active 
alkaloid  having  the  formula,  CwH^^O,  be  added  to  the  racemic 
modification  of  tartaric  acid,  the  cinchonine  salt  of  the  levo-acid 
will  crystallize  first.  The  crystals  of  the  cinchonine  salt  are  then 
removed  and,  after  adding  ammonia  to  displace  the  alkaloid, 


118  THEORETICAL  CHEMISTRY 

dilute  sulphuric  acid  is  added  and  the  pure  levo-tartaric  acid  is 
obtained. 

Methods  of  Ferments.  Notwithstanding  the  fact  that  optical 
antipodes  resemble  each  other  so  closely  in  most  of  their  properties, 
Pasteur  found  that  certain  micro-organisms  have  the  power  of 
distinguishing  sharply  between  these  forms.  For  example,  if 
penicillium  glaucum  be  introduced  into  a  solution  of  racemic 
tartaric  acid,  it  thrives  at  the  expense  of  the  dextro-acid  and 
eventually  leaves  the  pure  levo-form.  In  this  method,  one  of  the 
active  modifications  is  always  lost. 

Pasteur  was  the  first  to  point  out  that  there  must  be  some  inti- 
mate connection  between  optical  activity  and  the  constitution  of  the 
molecule.  It  remained  for  Le  Bel*  and  van't  Hoff ,  f  however,  to 
formulate  independently  and  almost  simultaneously  an  hypothesis 
to  account  for  optical  activity  on  the  basis  of  molecular  constitu- 
tion. Their  important  work  laid  the  foundation  of  spatial  chemis- 
try, commonly  termed  stereochemistry.  Le  Bel  accepted  Pasteur's 
view  that  optical  activity  is  dependent  upon  a  condition  of  asym- 
metry, but  whether  this  asymmetry  is  a  property  of  the  crystal 
alone,  or  whether  it  belongs  to  the  molecule  of  the  optically  active 
substance,  was  the  question  he  set  himself  to  answer.  He  found,  on 
dissolving  certain  optically  active  crystals  in  an  inactive  solvent, 
that  the  optical  activity  is  imparted  to  the  solution  and  therefore  he 
concluded  that  the  condition  of  asymmetry  must  exist  in  the  chem- 
ical molecule.  All  of  the  optically  active  substances  known  to  Le 
Bel  were  compounds  of  carbon.  An  examination  of  the  formulas 
of  these  compounds  led  him  to  ascribe  the  cause  of  their  optical 
activity  to  the  presence  of  an  asymmetric  carbon  atom,  that  is,  a  car- 
bon atom  combined  with  four  different  atoms  or  groups  of  atoms. 
One  of  the  simplest  examples  is  afforded  by  lactic  acid,  the  struc- 
tural formula  of  which  is 

H 

CH3  —  C  —  COOH 

I 
OH 

In  this  formula  the  asymmetric  carbon  atom  is  placed  at  the 
center  and  is  in  combination  with  hydrogen,  hydroxyl,  methyl 

*  Bull.  Soc.  Chim.  (2)  22,  337  (1874). 
t  Ibid.  (2),  23,  295  (1875). 


RELATION  BETWEEN  PHYSICAL  PROPERTIES 


119 


and  carboxyl.  In  connection  with  his  work  on  the  relation 
between  optical  activity  and  asymmetry,  Le  Bel  pointed  out  that 
active  forms  never  result  from  laboratory  syntheses,  the  racemic 
modification  being  invariably  obtained.  Van't  Hoff  reached  con- 
clusions similar  to  those  of  Le  Bel  and  proposed  the  additional 
theory  of  the  asymmetric  tetrahedral  carbon  atom.  Since  the  four 
valences  of  the  carbon  atom  are  equivalent,  as  the  work  of  Henry 
on  methane  has  shown  them  to  be,  van't  Hoff  pointed  out  that 
the  only  possible  geometrical  arrangement  of  the  atoms  in  the 
molecule  of  methane  must  be  that  in  which  the  carbon  atom  is 
placed  at  the  center  of  a  regular  tetrahedron  with  the  four  hy- 
drogen atoms  at  the  four  apices.  He  also  pointed  out  that 
when  the  four  valences  of  the  tetrahedral  carbon  atom  are  satis- 
fied with  different  atoms  or  groups,  no  plane  of  symmetry  can  be 
passed  through  the  figure,  the  carbon  atom  being  asymmetric. 
This  conception  of  Le  Bel  and  van't  Hoff  forms  the  basis  of 


CH 


OH    HO 


Fig.  38 


CO  OH 


CO  OH 


stereochemistry,  and  has  proved  of  inestimable  value  to  the 
organic  chemist  in  enabling  him  to  explain  the  existence  of  many 
isomeric  compounds.  Thus,  ordinary  lactic  acid  can  be  split  into 
two  optically  active  isomers.  Aside  from  the  fact  that  one  acid 
is  dextro-  and  the  other  is  levo-rotatory,  the  properties  of  the 
two  acids  are  practically  identical.  If  the  formulas  are  written 
spatially,  the  different  groups  can  be  arranged  about  the  asym- 
metric carbon  atom  in  such  a  way  that  the  two  tetrahedra  shall 
be  mirror  images  of  each  other,  as  shown  in  Fig.  38.  It  will 
be  observed  that  these  two  tetrahedra  can  in  no  way  be  super- 
posed so  that  the  same  groups  fall  over  each  other,  that  is  to 
say,  they  are  enantiomorphous  forms.  In  tartaric  acid  there  are 
two  asymmetric  carbon  atoms  as  is  evident  when  its  structural 
formula  is  written  as  follows :  — 


120 


THEORETICAL  CHEMISTRY 


H       H 
HOOC  —  C  —  C  —  COOH 

OH   OH 

If  the  stereochemical  formulas  of  the  dextro-  and  levo-acids  be 
represented  as  in  Fig.  39,  (a)  and  (b),  it  will  be  apparent  that 
the  theory  admits  of  the  existence  of  another  isomer  with  the 
atoms  and  groups  arranged  as  in  Fig.  39  (c). 

In  this  arrangement  the  asymmetry  of  the  upper  tetrahedron 
is  the  reverse  of  that  of  the  lower,  and  consequently  the  optical 
activity  of  one-half  of  the  molecule  exactly  compensates  the  optical 
activity  of  the  other  half,  and  the  molecule  as  a  whole  is  inactive. 
It  is  evident  that  such  a  tartaric  acid  could  not  be  split  into  two 
active  forms.  Actually  there  are  four  tartaric  acids  known,  viz., 


Racemic  Acid 
A 


COOH 

d-TartaricAcW 
«*) 


OH 


COOH 
Z-Tartaric  Acid 


OH 


CO  OH 

Meso-Tartaric  Acid 
(C) 


Fig.  39 


(1)  inactive  racemic  acid  which  is  separable  into  (2)  dextro-tar- 
taric  acid  and  (3)  levo-tartaric  acid;  and  (4)  meso-tartaric  acid, 
an  inactive  substance  which  has  never  been  separated  into  two 
active  forms,  but  which  has  the  same  formula,  the  same  molecular 
weight  and  in  general  the  same  properties  as  the  dextro-  or  levo- 
tartaric  acids.  Inactive  forms,  such  as  meso-tartaric  acid,  are  said 
to  be  inactive  by  internal  compensation.  This  constitutes  one  of 
many  beautiful  confirmations  of  the  van't  Hoff  theory  of  the 
asymmetric  tetrahedral  carbon  atom. 

Meso-tartaric  acid  furnishes  an  illustration  of  the  fact  that 
asymmetric  carbon  atoms  may  be  present  in  the  molecule  with- 
out imparting  optical  activity  to  the  substance.  The  converse 


RELATION  BETWEEN  PHYSICAL  PROPERTIES  121 

of  this  proposition,  however,  that  optical  activity  is  dependent 
upon  asymmetric  carbon  atoms,  is  generally  true.  Certain  sub- 
stances which  apparently  contain  no  asymmetric  carbon  atoms 
have  been  discovered  which  are,  nevertheless,  optically  active. 
An  example  of  such  a  substance  is  1  -methyl  cyclohexylidene-4 
acetic  acid,  to  which  the  following  formula  has  been  assigned;  — 


CH3CH  C  :  CH  -  COOH 

\          / 
CH2  •  CH2 

Other  atoms  aside  from  carbon  may  be  asymmetric;  thus  certain 
compounds  of  nitrogen,  sulphur  and  tin  have  been  shown  to  be 
optically  active.  The  theory  also  furnishes  an  explanation  of 
the  fact,  pointed  out  by  Le  Bel,  that  optically  active  forms  are 
never  obtained  by  direct  synthesis.  Since  the  rotatory  power  is 
dependent  upon  the  arrangement  of  the  atoms  and  groups  in  the 
molecule,  it  follows  from  the  doctrine  of  probability  that  as  many 
dextro  as  levo  configurations  will  be  formed,  and  consequently  the 
racemic  modification  will  be  obtained. 

Up  to  the  present  time  no  satisfactory  generalization  has  been 
discovered  as  to  the  factors  determining  the  molecular  rotation 
in  any  particular  case.  An  attempt  in  this  direction  has  been 
made  by  Guye,*  in  which  he  ascribes  the  magnitude  of  the  ob- 
served rotation  to  the  relative  masses  of  the  atoms  or  groups 
which  are  in  combination  with  the  tetrahedral  carbon  atom. 
But  it  cannot  be  mass  alone  which  conditions  optical  activity, 
since  substances  are  known  which  rotate  the  plane  of  polariza- 
tion, notwithstanding  the  fact  that  their  molecules  have  two 
groups  of  equal  mass  in  combination  with  the  asymmetric  car- 
bon atom.  The  molecular  rotations  of  the  members  of  homol- 
ogous series  exhibit  some  regularities,  but  on  the  other  hand 
many  exceptions  occur  which  cannot  be  satisfactorily  explained. 
About  all  that  can  be  said  at  the  present  time  is,  that  optical 
activity  is  a  constitutive  property. 

Magnetic  Rotation.  That  many  substances  acquire  the  power 
of  rotating  the  plane  of  polarized  light  when  placed  in  an  intense 
magnetic  field  was  first  observed  by  Faraday  f  in  1846. 

*  Compt.  rend.,  no,  714  (1890).  " 

t  Phil.  Trans.,  136,  1  (1846). 


122 


THEORETICAL  CHEMISTRY 


The  relation  between  chemical  composition  and  magnetic 
rotatory  power  has  since  been  investigated  very  exhaustively  by 
W.  H.  Perkin,*  his  experiments  in  this  field  having  been  con- 
tinued for  more  than  fifteen  years.  In  brief,  Perkin's  method  of 
investigating  magnetic  rotatory  power  consisted  in  introducing 
the  liquid  to  be  examined  into  a  polarimeter  tube  1  decimeter  in 
length  and  then  placing  the  tube  axially  between  the  perforated 
poles  of  a  powerful  electromagnet,  as  shown  in  Fig.  40.  Upon 
exciting  the  magnet  it  was  found  that  the  plane  of  polarization 
was  rotated,  either  to  the  right  or  the  left,  the  direction  of  rota- 
tion depending  upon  the  direction  of 
the  current,  the  intensity  of  the  mag- 
netic field  and  the  nature  of  the  liquid. 
Perkin  used  the  sodium  flame  as  his 
source  of  light  and  carried  out  all  of  his 
experiments  at  15°  C.  He  expressed 
his  results  by  means  of  the  formula, 
Ma/d,  a  being  the  observed  angle  of 
rotation,  d  the  density  of  the  liquid  and 


Fig.  40 


M  its  molecular  weight.  All  measurements  were  expressed  in 
terms  of  water  as  a  standard:  thus  if  Ma/d  is  the  rotation  for 
any  substance,  and  M'a'/d'  is  the  corresponding  rotation  for 
water,  then,  according  to  Perkin,  the  molecular  magnetic  rotation 
will  be  given  by  the  ratio,  Ma/d:  M'a  /d',  or  Mad'  /M'a'd. 

The  molecular  magnetic  rotation  for  a  large  number  of  organic 
compounds  has  been  determined  by  Perkin,  who  has  shown  it  to 
be  an  additive  property.  In  any  one  homologous  series  the  value 
of  the  molecular  magnetic  rotation  is  given  by  the  formula, 


mol.  mag.  rotation  =  a  +  nb, 


(19) 


where  a  is  a  constant  characteristic  of  the  series,  b  is  a  constant 
corresponding  to  a  difference  of  CH2  in  composition,  its  value 
being  1.023,  and  n  is  the  number  of  carbon  atoms  contained  in 
the  molecule.  This  formula  is  applicable  only  to  compounds 
which  are  strictly  homologous,  isomeric  substances  in  two  differ- 
ent series  giving  quite  different  rotations.  The  constitution  of 
the  molecule  exerts  as  great  an  influence  on  magnetic  rotation 

*  Jour,  prakt.  Chem.  [2],  31,  481  (1885);  Jour.  Chem.  Soc.,  49,  777;  41, 
808;  53,  561,  695;  59,  981;  61,  287,  800;  63,  57;  65,  402,  815;  67,  255;  69, 
1025  (1886-1896). 


RELATION  BETWEEN  PHYSICAL  PROPERTIES          123 

as  it  does  on  refraction,  a  double  bond  causing  an  appreciable 
increase  in  the  value  of  a.  The  results  of  experiments  on  mag- 
netic rotation  show  that  nothing  like  the  same  regularities  exist 
as  have  been  discovered  for  molecular  refraction  and  molecular 
volume.  The  rotatory  powers  of  various  inorganic  substances 
have  been  determined,  but  the  results  are  too  irregular  to  admit 
of  any  satisfactory  interpretation. 

Absorption  Spectra.  When  a  beam  of  white  light  is  passed 
through  a  colored  liquid  or  solution,  and  the  emergent  beam  is 
examined  with  a  spectroscope,  a  continuous  spectrum  crossed  by 
a  number  of  dark  bands  is  obtained.  A  portion  of  the  light  has 
been  absorbed  by  the  liquid.  Such  a  spectrum  is  known  as  an 
absorption  spectrum.  If  instead  of  passing  the  light  through  a 
liquid,  it  is  passed  through  an  incandescent  gas,  a  spectrum  will 
be  obtained  which  is  crossed  by  numerous  fine  lines,  termed 
Fraunhofer  lines.  Such  lines  occupy  the  same  positions  as  the 
corresponding  colored  lines  in  the  emission  spectrum  of  the  gas. 
It  follows,  therefore,  that  the  absorption  spectrum  is  quite  as  char- 
acteristic of  a  substance  as  its  emission  spectrum,  and  from  a 
careful  study  of  the  absorption  spectra  of  liquids  we  may  expect 
to  gain  some  insight  into  their  molecular  constitution.  The 
pioneer  workers  in  this  field  were  Hartley  and  Baly,*  and  it  is 
largely  to  them  that  we  owe  our  present  experimental  methods. 
The  instrument  employed  for  photographing  spectra  is  called  a 
spectrograph,  a  very  satisfactory  form  being  shown  in  Fig.  41. 
It  differs  from  an  ordinary  spectroscope  in  that  the  eye-piece  is 
replaced  by  a  photographic  camera.  This  attachment  is  clearly 
shown  in  the  illustration.  The  plateholder  is  so  constructed  that 
only  a,  narrow  horizontal  strip  of  the  plate  is  exposed  at  any  one 
time,  thus  making  it  possible  to  take  a  series  of  photographs  on 
the  same  plate  by  simply  lowering  the  holder.  By  means  of  a 
millimeter  scale,  also  shown  in  the  illustration,  the  plateholder 
can  be  moved  through  the  same  distance  each  time  before  expos- 
ing a  fresh  portion  of  the  plate,  thus  insuring  an  equally-spaced 
series  of  spectrum  photographs.  In  order  that  spectra  in  the 
ultra-violet  region  may  be  photographed,  it  is  customary  to  equip 
the  instrument  with  quartz  lenses  and  a  quartz  prism,  ordinary 
glass  not  being  transparent  to  the  ultra-violet  rays.  Using  a 

*  See  numerous  papers  in  the  JQUI.  Chem.  Soc.,  since  1880. 


124 


THEORETICAL  CHEMISTRY 


spectrograph  furnished  with  a  quartz  optical  system,  it  is  possible 
to  photograph,  on  a  single  plate,  the  entire  spectrum  from  2000  to 
8000  Angstrom  units.  A  scale  of  wave-lengths  photographed  on 
glass  is  provided  with  the  instrument,  so  that  the  wave-lengths  of 
lines  or  bands  can  be  read  off  directly  by  laying  the  scale  over  the 
photographs. 


Fig.  41 

The  source  of  light  to  be  used  depends  upon  the  character  of 
the  investigation.  If  a  source  rich  in  ultra-violet  rays  is  desired, 
the  light  from  the  electric  spark  obtained  between  electrodes  pre- 
pared from  an  alloy  of  cadmium,  lead  and  tin  is  very  satisfactory; 
or  the  light  from  an  arc  burning  between  iron  electrodes  may  be 
used.  For  investigations  in  the  visible  region  of  the  spectrum 
the  Nernst  lamp  is  unsurpassed.  In  using  the  spectrograph  for 
the  purpose  of  studying  the  constitution  of  a  dissolved  substance* 
it  is  necessary  to  determine  not  only  the  number  and  position  of 
the  absorption  bands,  but  also  the  persistence  of  these  bands  as 
the  solution  is  diluted. 

According  to  Beer's  law  the  product  of  the  thickness,  t,  of  an 
absorbing  layer  of  solution  of  molecular  concentration,  m,  is  con- 
stant, or  mt  =  k.  If  then,  the  thickness  of  a  given  layer  of  solu- 


RELATION  BETWEEN  PHYSICAL  PROPERTIES 


125 


tion  is  diminished  n  times,  its  absorption  will  be  the  same  as  that 
of  a  solution  whose  concentration  is  only  1  /nth  of  that  of  the 
original  solution.  Thus,  by  varying  the  thickness  of  the  absorbing 
layer  we  can  produce  the  same  effect  as  by  changing  the  concen- 
tration. The  convenient  device  of  Baly  for  altering  the  length 
of  the  absorbing  column  of  liquid  is  shown  in  Fig.  42,  attached  to 
the  collimator  of  the  spectrograph.  It  consists  of  two  closely- 
fitting  tubes,  one  end  of  each  tube  being  closed  by  a  plane,  quartz 
disc.  The  outer  tube  is  fitted  with  a  small  bulbed-funnel  and  is 


Fig.  42 

graduated  in  millimeters.  The  two  tubes  are  joined  by  means  of 
a  piece  of  rubber  tubing  which  prevents  leakage  of  the  contents, 
and  at  the  same  time  admits  of  the  adjustment  of  the  column  of 
liquid  to  the  desired  length  by  simply  sliding  the  smaller  tube  in 
or  out. 

Molecular  Vibration  and  Chemical  Constitution.  There  are 
two  systems  of  graphic  representation  of  the  results  of  spectro- 
scopic  investigations.  In  the  first  system,  due  to  Hartley,  the 
wave-lengths  or  their  reciprocals,  the  frequencies,  are  plotted  as 
abscissae,  and  the  thicknesses,  in  millimeters,  of  the  absorbing 
layers,  are  plotted  as  ordinates.  Such  curves  are  known  as 
curves  of  molecular  vibration.  The  second  system,  due  to  Baly 
and  Desch,  is  a  modification  of  that  developed  by  Hartley. 
Baly  and  Desch  suggested  that,  for  various  reasons,  it  would  be 
more  advantageous,  if  instead  of  plotting  the  thickness  of  the 
absorbing  layers  as  ordinates,  the  logarithms  of  these  thicknesses 
be  plotted.  Both  methods  have  their  advantages  and  both  are 


126 


THEORETICAL  CHEMISTRY 


used.  As  an  illustration  of  the  value  of  curves  of  molecular 
vibration  in  connection  with  questions  of  chemical  constitution, 
we  will  take  the  case  of  o-hydroxy-carbanil.  The  constitution 
of  this  substance  was  known  to  be  represented  by  one  of  the  two 
following  formulas  :  — 


NH 


O 


or 


CeH 


-OH 


(a) 


(b) 


On  comparing  the  curves  of  molecular  vibration  for  the  three  sub- 
stances (Fig.  43),  it  is  apparent  that  the  curves  for  the  lactam 


4000 


Oscillation  Frequencies 

4000 
...III. 


4000 


V.. 


ttl 


Ltictn.ni  J3tlicr 


O-Hydcoxy-Carbanil 
Fig.  43 


Lactim  Ether 


ether  and  o-hydroxy-carbanil  bear  a  close  resemblance  to  each 
other,  while  the  curve  for  the  lactim  ether  is  very  different  from 
the  curves  for  the  other  two  substances.  The  constitution  of 


RELATION  BETWEEN  PHYSICAL  PROPERTIES  127 

o-hydroxy-carbanil  must  then  be  very  similar  to  that  of  the 
lactam  ether.  The  formulas  of  the  ethyl  derivatives  of  the 
mother  substance  are  known  to  be  as  follows  :  — 


-  OC2H6 

-  C2H5  X 

Lactam  ether  Lactim  ether 

Hartley  concluded,  therefore,  that  formula  (a)  represents  the 
structure  of  the  molecule  of  o-hydroxy-carbanil. 

It  is  beyond  the  scope  of  this  book  to  discuss  at  greater  length 
the  bearing  of  absorption  spectra  upon  chemical  constitution; 
but  the  student  is  earnestly  advised  to  consult  spme  book  treating 
of  this  important  subject,  or  to  read  some  of  the  original  papers. 

Dielectric  Constants  and  Absorption  of  Electric  Vibrations. 
Faraday  discovered  that  the  attraction  or  repulsion  between  two 
electric  charges  varies  with  the  nature  of  the  intervening  medium 
or  dielectric.  If  qi  and  g2  represent  two  charges  which  are  sepa- 
rated by  a  distance  r,  the  force  of  attraction  or  repulsion,  /,  is 
given  by  the  equation 


where  D  is  a  specific  property  of  the  medium  known  as  the  dielectric 
constant.  The  dielectric  constant  of  air  is  taken  as  unity.  Vari- 
ous methods  have  'been  devised  for  the  experimental  determina- 
tion of  the  dielectric  constant,  but  the  scope  of  this  book  forbids 
a  description  of  the  apparatus  or  an  outline  of  the  processes  of 
measurement.  For  a  description  of  these  methods  the  student 
is  referred  to  any  one  of  the  more  complete  physico-chemical 
laboratory  manuals.* 

According  to  the  electro-magnetic  theory  of  light, 

D  =  n2,  (21) 

*  Until  recently  there  has  been  no  really  satisfactory  method  available  for 
the  accurate  measurement  of  the  dielectric  constants  of  liquids  possessing  a 
specific  conductivity  greater  than  that  of  so-called  conductivity  water.  Pro- 
fessor W.  A.  Patrick  has  devised  a  method  by  which  such  measurements  may 
be  made  with  a  high  degree  of  accuracy  and  we  may  anticipate  important 
results  in  this  interesting  field.  For  a  description  of  the  apparatus  and 
method  of  measurement  the  student  is  referred  to  Prof.  Patrick's  paper, 
Jour.  Am.  Chem.  Soc.,  43,  1835  (1921). 


128  THEORETICAL  CHEMISTRY 

• 

where  D  is  the  dielectric  constant  of  an  insulating  medium  and  n 
is  the  index  of  refraction  for  electric  vibrations  of  infinite  wave- 
length. Notwithstanding  the  fact  that  the  foregoing  relation 
is  theoretically  valid  for  vibrations  of  infinite  wave-length  only, 
it  has  been  shown  by  experiment  to  hold  very  satisfactorily  for 
vibrations  whose  wave-lengths  range  from  one  to  two  meters. 
With  substances  possessing  moderate  electric  conductivity,  it 
has  been  found,  on  measuring  the  index  of  refraction  for  electric 
vibrations  of  finite  wave-length,  that  the  calculated  values  of  D 
are  not  satisfactory.  Under  these  conditions,  it  is  necessary  to 
make  allowance  for  the  absorbing  power  of  the  medium  and 
equation  (21)  becomes 

(  D  =  n2  (1  -  xz),  (22) 

where  x  represents  the  absorption  index.  In  order  that  the 
value  of  x  may  be  calculated,  the  conductivity  and  the  dielectric 
constant  of  the  medium  and  the  wave-length  of  the  incident 
electric  vibrations  must  -  be  known.  With  normal  substances, 
the  absorbing  power  increases  with  increase  in  both  the  con- 
ductivity of  the  medium  and  the  wave-length  of  the  incident 
vibrations.  Since  most  organic  compounds  are  poor  conductors 
of  electricity,  they  are  found  to  be  practically  non-absorbers  of 
all  electric  vibrations  of  wave-lengths  under  one  meter.  There 
are  some  organic  compounds,  however,  which  behave  exceptionally 
in  this  respect,  and,  notwithstanding  the  fact  that  they  are  prac- 
tically non-conductors,  they  exhibit  marked  absorbing  power  for 
electric  vibrations  of  comparatively  short  wave-lengths.  With 
these  anomalous  absorbers,  the  value  of  n2  is  invariably  less  than 
the  dielectric  constant  of  the  medium,  and  the  absorbing  power 
increases  with  decreasing  wave-length  of  the  incident  vibrations. 

Drude  *  was  the  first  to  point  out  that,  with  the  exception  of 
water,  all  compounds  containing  hydroxyl  groups  exhibit  anoma- 
lous electric  absorption.  He  also  showed  that  anomalous  electric 
absorption  is  conditioned  by  the  dielectric  constant  of  the  medium, 
those  compounds  which  have  small  dielectric  constants  being 
characterized  by  small  absorbing  power.  While  Drude  inferred 
that  compounds  which  do  not  contain  hydroxyl  groups  are  non- 
absorbers,  subsequent  investigations  render  it  highly  probable 
that  there  are  other  groups  beside  the  hydroxyl  group  which  favor 

*  Wied.  Ann.,  55,  633  (1895);  58,  1  (1896);  59,  17  (1896);  60,  50  (1897); 
Zeit.  phys.  Chem.,  23,  282  (1897). 


RELATION  BETWEEN  PHYSICAL  PROPERTIES 


129 


anomalous  electric  absorption.     In  general  it  may  be  said  that 
anomalous  electric  absorption  is  a  constitutive  property. 

The  values  of  the  dielectric  constants  for  some  of  the  more 
common  solvents  are  given  in  the  accompanying  table. 

DIELECTRIC  CONSTANTS  AT   18°  C. 


Substance 

D 

Hydrogen  dioxide 

92  8 

Water  

77  0 

Formic  acid 

63  0 

Methyl  alcohol  ... 

33  7 

Ethyl  alcohol  

25.8 

Ammonia,  liquid  
Chloroform  

22.0 
5.0 

Ether 

4  35 

Carbon  tetrachloride 

2  25 

Benzene  

2.28 

The  importance  of  this  property  will  become  more  apparent  in 
subsequent  chapters,  especially  in  those  devoted  to  electrochem- 
istry. 

REFERENCES 

The  molecular  Volumes  of  Liquid  Chemical  Compounds.     Gervaise  Le  Bas. 
The  Relations  between  Chemical  Constitution  and  Some  Physical  Proper- 
ties.    Samuel  Smiles'. 

Optical  Rotation  of  Organic  Substances.     Landolt.     (Translation  by  J.  H. 
Long.) 

PROBLEMS 

1.  Calculate  the  molecular  volume  of  ethyl  butyrate.    The  molecular 
volume  determined  by  experiment  is  149.1. 

2.  For  propionic  acid,  d  =  1.0158  and  nD  =  1.3953.     Calculate  the 
molecular  refraction  by  the  formula  of  Lorenz-Lorentz  and  compare  the 
value  so  obtained  with  that  derived  from  the  atomic  refractions  of  the 
constituent  elements. 

3.  The  density  of  ether  is  0.7208,  of  ethyl  alcohol,  0.7935  and  of  a 
mixture  of  ether  and  alcohol  containing  p  per  cent  of  the  latter,  0.7389. 
At  20°  C.  the  refractive  indices  for  sodium  light  are,  for  ether,  1.3536, 
for  alcohol,  1.3619,  and  for  the  mixture,  1.3572.     Calculate  the  value  of 
p,  using  the  Gladstone  and  Dale  formula.  Ans.  20.81. 

4.  At  20°  C.  the  density  of  chloroform  is  1.4823  and  the  refractive 
index  for  the  D-line  is  1.4472.     Given  the  atomic  refractivities  of  carbon 
and   hydrogen,  calculate  that  of  chlorine,  using  the   Lorenz-Jibrentz 
formula. 


CHAPTER  VI 

ELEMENTARY  PRINCIPLES   OF  THERMO- 
DYNAMICS 

Thermodynamics  Defined.  The  science  of  thermodynamics 
treats  of  the. laws  governing  the  quantitative  transformations  of 
heat  energy  into  other  forms  of  energy.  In  other  words,  it  is  the 
science  of  heat  considered  as  a  form  of  energy,  and  includes  the 
study  of  the  relation  of  heat  energy  to  all  other  forms  of  energy 
whether  mechanical,  electrical  or  chemical.  A  knowledge  of  the 
fundamental  principles  of  thermodynamics  is  essential  to  the 
student  of  physical  chemistry;  in  fact,  thermodynamics  consti- 
tutes the  underlying  framework  of  the  entire  science  of  physical 
chemistry. 

Energy.  Since  thermodynamics  deals  with  energy  and  its 
transformations  it  is  essential  that  the  student  should  have  a 
clear  understanding  of  the  meaning  of  the  term  "  energy."  En- 
ergy may  be  defined  as  the  capacity  of  a  body,  or  system,  to  perform 
work.  Since  the  work  done  affords  a  measure  of  the  energy  ex- 
pended, it  follows  that  both  energy  and  work  are  to  be  measured 
in  terms  of  the  same  unit,  the  erg.  Energy  stored  in  a  system 
by  virtue  of  its  motion  is  known  as  kinetic  energy.  This  defini- 
tion applies  equally  to  the  motion  of  the  molecules  constituting 
a  system  and  to  the  motion  of  the  system  as  a  whole.  Energy 
stored  in  a  system  by  virtue  of  its  position  is  called  potential 
energy.  The  energy  contained  in  a  body  by  virtue  of  its  tem- 
perature and  molecular  structure,  or  the  sum  of  its  internal  kinetic 
and  potential  energies,  is  known  as  its  intrinsic  energy.  Since  the 
deductions  of  thermodynamics  are  independent  of  any  molecular 
hypothesis,  we  are  not  so  much  concerned  as  to  whether  the 
energy  of  a  system  is  kinetic  or  potential,  as  we  are  to  ascertain 
the  gain  or  loss  in  energy  which  the  system  may  experience  in 
undergoing  a  given  transformation. 

Although  we  may  employ  algebraic  symbols  to  represent  the 
total  energy  stored  in  a  system,  it  should  be  borne  in  mind  that  we 

130 


ELEMENTARY  PRINCIPLES  OF  THERMODYNAMICS     131 

possess  no  means  of  measuring  this  energy.  When  a  process 
occurs  in  which  the  energy  of  a  system  undergoes  change,  our 
measurements  merely  give  us  the  gain  or  loss  in  the  energy  of 
the  system  as  a  result  of  the  change.  For  example,  let  a 
body  having  a  mass  of  one  gram  fall,  under  the  influence  of 
gravity,  through  a  distance  of  one  meter:  the  resulting  loss  in 
potential  energy  is  1  g.  X  981  cm./sec.2  X  1000  cm.  =  981,000 
ergs.  In  this  calculation,  the  absolute  values  of  the  potential 
energy  of  the  body  in  its  initial  and  final  states  are  not  involved, 
and  all  that  we  have  learned  is,  that  in  its  final  state  the  body 
possesses  981,000  ergs  less  potential  energy  than  it  did  in  its  ini- 
tial state. 

All  forms  of  energy  may  be  expressed  as  the  product  of  two 
factors,  —  an  "  intensity  factor  "  and  a  "  capacity  factor."  Thus, 
in  the  preceding  example,  potential  energy  is  the  product  of  a 
force,  —  the  intensity  factor,  —  and  a  distance,  —  the  capacity 
factor:  or  to  cite  another  example,  heat  energy  is  the  product  of 
temperature,  —  the  intensity  factor,  —  and  quantity  of  heat,  — 
the  capacity  factor. 

The  First  Law  of  Thermodynamics.  The  science  of  thermo- 
dynamics is  based  upon  two  fundamental  principles  known  as 
the  first  and  second  laws  of  thermodynamics.*  While  the  first 
law  is  a  generalization  derived  from  direct  experiments,  the  sec- 
ond law  rests  for  its  verification  upon  human  experience:  though 
less  rigorously  established,  the  acceptance  of  the  second  law  is  justi- 
fied by  the  fact  that  no  exceptions  to  it  have  as  yet  been  found. 

The  first  law  is  merely  a  statement  of  the  principle  of  the  con- 
servation of  energy,  according  to  which  we  learn  that  energy  may 
be  changed  from  one  form  to  another,  but  that  it  can  never  be 
created  or  destroyed.  This  fundamental  principle  was  first  clearly 
formulated  by  Mayer  in  1842,  shortly  after  the  publication  of  the 
results  of  the  important  investigations  of  Helmholtz,  Joule,  Hirn 
and  others  on  the  transformation  of  heat  into  work  and  work  into 
heat.  These  experimenters  established  the  fact  that  when  work  is 
transformed  completely  into  heat,  or  heat  into  work,  the  quantity  of 
work  is  equivalent  to  the  quantity  of  heat.  Thus,  if  Q  units  of  heat 
disappear  in  any  process,  and  W  units  of  work  are  produced,  then, 

*  Another  principle,  originally  advanced  by  Nernst  and  since  confirmed 
by  others,  is  now  commonly  referred  to  as  the  "  Third  Law  of  Thermody- 
namics." Its  consideration  lies  beyond  the  scope  of  this  book. 


P 


132  THEORETICAL  CHEMISTRY 

if  no  other  forms  of  energy  are  involved,  the  following  relation 
holds :  — 

W  =  JQ,  (1) 

where  J  is  a  proportionality  factor,  called  the  mechanical  equiva- 
lent of  heat.  The  value  of  the  mechanical  equivalent  of  heat  as 
computed  from  the  most  reliable  sources  is  J  =  4.183  X  107  ergs 
per  gram-calorie. 

In  order  that  we  may  derive  a  general  mathemat- 
ical expression  of  the  first  law,  let  us  imagine  v  cc. 
of  gas  to  be  enclosed  in  a  cylinder  which  is  fitted 
with  a  frictionless,  weightless  piston,  as  shown  in 
Fig.  44.  If  the  temperature  be  raised  a  small 
amount,  Q  units  of  heat  will  be  imparted  to  the 
gas,  the  total  or  internal  energy  of  the  system  will 
be  increased  by  U  units,  and  the  system  will  also 
44  perform  W  units  of  external  work  against  the 
atmosphere.  Therefore,  according  to  the  principle 
of  the  conservation  of  energy,  we  write 

U  =  Q  -  W.  (2) 

This  is  a  perfectly  general  mathematical  formulation  of  the  first 
law  of  thermodynamics. 

If  in  the  foregoing  process,  the  volume  of  the  gas  increases  by 
a  very  small  amount,  dv,  the  external  work  done  will  be  equal  to 
pdv,  and  equation  (2)  may  be  written  in  the  form 

U  =  Q  -  pdv.  (3) 

In  the  practical  application  of  this  and  other  thermodynamic 
equations,  it  is  very  important  to  keep  in  mind  that  all  of  the 
quantities  involved  shall  be  expressed  in  terms  of  the  same  unit 
of  energy.  The  signs  of  the  terms  in  equation  (2)  must  be  deter- 
mined by  the  nature  of  the  process  to  which  the  equation  is  applied. 
It  is  customary  to  designate  increase  in  internal  energy  by  +  U, 
absorption  of  heat  by  +  Q  and  performance  of  external  work  on 
the  surroundings  by  +  W. 

If  a  system  undergoes  change  without  any  alteration  in  volume, 
it  is  apparent  that  no  external  work  will  be  performed,  either  by 
or  on  the  system,  and  consequently,  equation  (2)  reduces  to 

U  =  Q,,  (4) 

where  Qv  denotes  heat  absorbed  at  constant  volume. 


ELEMENTARY  PRINCIPLES  OF  THERMODYNAMICS      133 

It  is  important  to  observe  that  Q  does  not  afford  a  measure 
of  U.  Thus,  it  is  quite  possible  that  W  and  U  may  be  nearly, 
equal  numerically,  in  which  case  (W  —  U)  will  be  almost  zero, 
and  yet  Q  may,  at  the  same  time,  have  a  relatively  large  numerical 
value. 

Isothermal  Expansion.  When  a  gas  expands  its  temperature 
will  fall,  unless  an  amount  of  heat  is  supplied  from  its  surround- 
ings sufficient  to  compensate  for  the  energy  expended  in  the  per- 
formance of  external  work.  When  this  natural  tendency  of  a  gas 
to  undergo  cooling  during  free  expansion  is  thus  offset,  and  the 
temperature  of  the  gas  remains  constant,  the  gas  is  said  to  expand 
isothermally. 

Maximum  Work.  Let  us  consider  the  isothermal  expansion 
of  one  gram-molecule  of  a  perfect  gas  enclosed  in  a  cylinder  fitted 
with  a  frictionless  and  weightless  piston.  In  order  that  the  gas 
may  perform  a  maximum  amount  of  work  in  expanding  through 
an  infinitely  small  volume,  dv,  it  is  necessary  that  the  external 
pressure  on  the  piston  shall  be  less  than  the  pressure  of  the  gas, 
p,  by  an  infinitely  small  amount,  dp.  Under  these  conditions, 
the  effective  pressure  to  be  overcome  by  the  gas  in  expanding  is, 
(p  —  dp),  while  the  work  done  will  be  (p  —  dp)dv  =  pdv,  the  pro- 
duct dp.dv  being  an  infinitely  small  quantity  of  the  second  order. 
If  the  external  pressure  is  just  equal  to  the  gas  pressure,  obviously 
the  piston  will  remain  stationary,  and  no  work  will  be  done.  If, 
however,  the  value  of  the  external  pressure  be  increased  so  as  to 
exceed  the  pressure  of  the  gas  by  dp,  the  effective  pressure  on  the 
piston  will  be  (p  -f  dp)  and  an  amount  of  work  —  (p  +  dp)dv  = 
—  pdv,  will  be  done  upon  the  gas,  the  negative  sign  indicating 
that  the  volume  of  the  gas  is  decreasing.  It  follows,  therefore,  that 
the  numerical  value  of  the  work  performed  by  the  gas  expanding 
isothermally  against  a  pressure  (p  —  dp),  is  equal  to  the  numer- 
ical value  of  the  work  performed  upon  the  gas  by  a  pressure 
(p  -f-  dp).  Such  a  process  is  said  to  be  reversible. 

If  instead  of  expanding  through  an  infinitely  small  volume,  dv, 
the  gas  be  allowed  to  undergo  an  appreciable  expansion,  say  from 
volume  vi  to  volume  v*,  the  pressure  of  the  gas  cannot  be  taken 
as  a  constant  quantity,  since,  according  to  Boyle's  law,  the  pres- 
sure varies  inversely  as  the  volume.  The  expression  p(v%  —  vi), 
therefore,  no  longer  holds  for  the  maximum  work  done  by  the 
gas.  In  order  to  calculate  the  maximum  work  done  by  a  gas 


134  THEORETICAL  CHEMISTRY 

in  expanding  isothermally  through  a  measureable  volume,  the 
ytotal  expansion  from  volume  v\  to  volume  v2,  must  be  treated  as 
the  sum  of  a  series  of  infinitely  small  expansions  in  each  of  which 
the  increment  of  volume,  dv}  is  multiplied  by  the  value  of  the 
pressure  corresponding.  The  total  work,  W,  will  then  be 
expressed  by  the  equation, 

W  =  ppdv.  (5) 

Since  for  a  perfect  gas,  pv  =  RT,  or  p  =  RT/v,  we  have 

W=RTr^.  (6 

Jvi     V 

Integrating  equation  (6),  we  obtain 

*»« 

(7) 


But  Vi  =  RT/pi  and  v2  =  RT/p2,  hence  equation  (7)  takes  the 
form 

W  =  RT  \oge-.  (8) 

Equation  (7)  is  an  expression  for  the  maximum  work  performed  by 
one-gram  molecule  of  gas  in  expanding,  isothermally  and  reversibly, 
from  volume  v\  to  volume  v2.  Equation  (8)  is  a  similar  expression 
involving  pressure  instead  of  volume. 

Reversible  and  Irreversible  Processes.  Any  process  in  which 
the  successive  steps  of  the  direct  process  occur  in  exactly  the  re- 
verse order,  is  called  a  reversible  process.  A  reversible  process 
is  to  be  considered  as  the  ideal  limit  to  which  actual  processes 
may  approach  ever  so  closely,  but  never  completely  attain. 
Although  no  natural  process  is  known  in  which  strict  reversibil- 
ity can  be  fully  realized,  nevertheless,  thermodynamic  deductions 
based  upon  the  assumption  of  completely  reversible  processes 
are  perfectly  valid. 

An  irreversible  process  is  one  in  which  the  successive  steps  of 
the  direct  process  cannot  be  retraced.  In  such  a  process  the  sys- 
tem suffers  a  permanent  loss  of  energy,  in  consequence  of  which, 
the  amount  of  work  performed  never  attains  to  the  theoretical 
maximum  value. 


ELEMENTARY  PRINCIPLES  OF  THERMODYNAMICS     135 

Reversible  and  Irreversible  Cycles.  If  after  the  completion 
of  any  process,  a  system  is  restored  to  its  initial  state,  it  is  said 
to  have  undergone  a  cycle  of  changes,  and  the  process  is  referred 
to  as  a  cyclic  process.  Cycles  may  be  either  reversible,  or  irre- 
versible, according  to  whether  they  can  be  performed  in  the  re- 
verse order  or  not. 

Since  after  the  completion  of  a  reversible  cycle,  the  chemical 
and  physical  properties  of  a  system  are  restored  to  their  original 
values,  it  follows  that  the  internal,  or  total,  energy  of  the  system 
has  not  undergone  any  change.  It  should  be  pointed  out,  how- 
ever, that  although  the  value  of  the  internal  energy  of  the  system 
remains  unchanged,  heat  may  be  evolved  or  absorbed,  and  work 
may  be  done,  either  by,  or  upon  the  system  during  a  reversible 
cyclic  process.  Since  for  a  completely  reversible  cycle,  U  =  0,  it 
follows,  from  the  first  law  of  thermodynamics,  that  W  =  Q,  or, 
if  during  the  cycle  there  are  several  operations  in  which  exchanges 
of  heat  and  work  occur,  the  sum  of  all  the  terms  involving  work, 
(STF),  must  be  equal  to  the  sum  of  all  the  terms  involving  heat, 
(ZQ);  that  is,  STF  =  2Q.  This  equality  holds,  irrespective  of  the 
physical  state  of  the  system,  whether  the  cycle  be  isothermal  or 
nonisothermal.  Furthermore,  it  can  be  shown  that  the  work  done 
in  any  reversible,  isothermal,  cyclic  process  is  zero. 

In  the  application  of  thermodynamics  to  the  ordinary  prob- 
lems of  physics  and  chemistry,  reversible  cyclic  processes  are  by 
far  the  more  important. 

The  Gay-Lussac- Joule  Experiment.  In  order  to  determine 
whether  the  temperature  of  a  gas  undergoes  any  change  when 
the  gas  expands  into  a  vacuum,  Gay-Lussac,  in  1807,  carried  out 
the  following  important  experiment.  Two  large  vessels  of  equal 
capacity  and  provided  with  thermometers,  were  connected 
by  means  of  a  pipe,  fitted  with  a  valve.  One  vessel  was  filled 
with  gas,  and  the  other  was  evacuated.  On  opening  the  valve 
and  allowing  the  gas  to  expand  into  the  evacuated  vessel,  the 
temperature  of  the  gas  in  the  first  vessel  was  observed  to  fall. 
A  corresponding  rise  in  temperature  occurred  in  the  second  vessel, 
due  to  the  compression  of  its  contents.  When  equilibrium  of 
pressure  had  been  established,  the  decrease  of  temperature  in  the 
first  vessel  was  found  to  be  exactly  equal  to  the  increase  of  tem- 
perature in  the  second  vessel.  From  this,  Gay-Lussac  concluded 
that,  taken  as  a  whole,  the  gas  showed  no  tendency  to  become 


136  THEORETICAL  CHEMISTRY 

warmer,  or  colder,  during  free  expansion.  A  somewhat  similar 
x experiment  was  performed  by  Joule  in  1844.  He  immersed  two 
similarly  connected  vessels  in  a  calorimeter,  and  found  that  no 
appreciable  change  in  the  temperature  of  the  water  of  the  calo- 
rimeter occurred  during  the  free  expansion  of  a  gas. 

From  these  two  experiments  it  is  evident,  that  when  a  perfect 
gas  expands  freely  into  a  vacuum,  heat  is  neither  evolved  nor 
absorbed.  Furthermore,  the  volume  of  the  system,  as  a  whole, 
remains  constant  and,  therefore,  no  external  work  is  done.  Since 
Q  =  0  and  W  =  0,  in  the  equation,  U  =  Q  —  W,  it  follows  that 
U  =  0,  i.e.,  the  internal  energy  of  a  given  mass  of  a  gas  is  inde- 
pendent of  the  volume.  Subsequent  experiments  by  Joule  and 
Thomson  have  shown  that  this  statement  is  only  approxi- 
mately true  for  actual  gases.  A  slight  change  in  temperature 
does  occur  when  a  gas  expands  freely,  due  to  the  fact  that 
a  certain  amount  of  work  is  required  to  overcome  the  mutual 
attraction  of  the  molecules  for  each  other.  For  a  perfect  gas, 
however,  in  which  no  such  attractive  forces  exist,  the  internal 
energy  is  independent  of  the  volume.  This  phenomenon  is  known 
as  the  Joule-Thomson  effect. 

The  Second  Law  of  Thermodynamics.  Attention  has  already 
been  directed  to  the  fact,  that  although  work  can  always  be  trans- 
formed into  heat,  this  is  not  equally  true  of  the  reverse  process,  — 
the  transformation  of  heat  into  work.  The  first  law  of  thermo- 
dynamics merely  specifies,  that  when  heat  is  convertedin  to  work, 
a  definite  quantitative  relationship  exists  between  the  heat  ab- 
sorbed and  the  work  done,  but  it  tells  us  absolutely  nothing  as 
to  the  amount  of  work  which  can  be  obtained  from  a  given  quan- 
tity of  heat.  It  is  to  the  second  law  of  thermodynamics  that  we 
must  turn  for  information  as  to  the  limitations  which  experience 
has  shown  govern  the  transformation  of  heat  into  work.  The 
second  law,  as  stated  by  Clausius,  is  as  follows: —  "  It  is  impos- 
sible for  a  self-acting  machine,  unaided  by  any  external  agency,  to 
convey  heat  from  one  body  to  another  at  a  higher  temperature.  In 
other  words,  heat  cannot  of  itself  pass  from  a  colder  to  a  hotter 
body,  but  tends  invariably  toward  a  lower  thermal  level.  It 
should  be  clearly  understood  that  this  is  in  no  way  contradictory 
to  the  first  law.  The  first  law  is  concerned  solely  with  the  quan- 
titative relationship  governing  the  transformation  of  heat  into 


ELEMENTARY  PRINCIPLES  OF  THERMODYNAMICS     137 

work:  it  does  not  affirm  that  such  transformations  are  possible 
under  all  conditions. 

Mathematical  Formulation  of  the  Second  Law.  A  mathemat- 
ical expression  of  the  second  law  can  be  derived  from  a  consider- 
ation of  the  behavior  of  a  perfect  gas  undergoing  a  reversible 
cycle  of  changes.  Let  us  imagine  one  gram-molecule  of  a  per- 
fect gas,  the  volume  of  which  is  vi  at  the  temperature  T,  to  be 
enclosed  in  a  cylinder  fitted  with  a  frictionless  and  weightless 
piston.  Let  the  gas  expand  isothermally,  from  the  volume  vi 
to  the  volume  vz.  In  this  expansion,  the  maximum  work  which 
the  gas  can  perform  is,  according  to  equation  (7), 

W  =  RTlo&^-  (9) 

Since  the  change  in  internal  energy,  U,  of  a  perfect  gas  depends 
only  on  the  temperature,  it  follows,  that  U  =  0,  if  the  gas 
expands  isothermally.  According  to  the  first  law, 

U  =  Q  -  W, 

but  U  =  0  and  consequently  W  =  Q.  Substituting  this  value 
of  W  in  equation  (9),  we  have 


>  (10) 

or  l 


It  is  of  importance  to  observe  that  while  W,  in  the  foregoing  equa- 
tions, represents  the  maximum  work  which  can  be  done  by  a  gas  in 
expanding  isothermally  from  volume  Vi,  to  volume  t;2,  it  by  no 
means  implies  that  such  a  maximum  value  must  necessarily  be 
attained.  For  example,  in  the  experiment  of  Gay-Lussac  on  the 
expansion  of  a  gas  into  a  vacuum,  the  value  of  W  is  always  zero. 

The  value  of  RT\oge-  ,  on  the  other  hand,  is  a  maximum  value. 
Vi 

Thus,  if  we  compress  the  gas  isothermally,  until  the  original  volume 
vi,  is  restored,  the  work  involved,  under  ideal  conditions,  must, 
at  least,  be  equal  to  W.  Therefore,  the  maximum  work  obtain- 
able with  a  perfect  engine  must  be  that  corresponding  to  a  rever- 
sible process.  Let  the  foregoing  operations  be  repeated  at  a 
higher  temperature,  T  -+-  dT.  The  maximum  work  done  will  be 


W  +  dW  =  R(T  +  dT)\o&-.  (12) 


138  THEORETICAL  CHEMISTRY 

It  will  be  observed,  that  the  change  in  volume  is  the  same  as  in  the 
preceding  case,  although  the  temperature  is  higher.  On  sub- 
tracting equation  (10)  from  equation  (12),  we  obtain 

dW  =  RdTloge^'  (13) 

But  according  to  equation  (11), 

v^T' 
therefore,  we  have 

04£-  (14) 


This  is  the  mathematical  expression  of  the  second  law  of  thermo- 
dynamics. 

It  should  be  borne  in  mind,  that  equation  (14)  holds  only  for 
a  reversible  cycle.  If  the  cycle  is  irreversible,  the  amount  of 
work  obtainable  will  be  less  than  that  given  by  the  equation. 
Hence,  for  irreversible  processes  we  have  the  inequality, 


Q-  (15) 

Equation  (14)  can  be  transformed  as  follows:  — 


where  the  partial  differential  coefficient  denotes  the  rate  of  change 
in  the  maximum  work  with  the  absolute  temperature,  at  constant 
volume.  This  is  a  convenient  form  in  which  to  use  the  equation 
expressing  the  second  law  of  thermodynamics. 

The  Gibbs-Helmholtz  Equation.  In  order  that  a  system  may 
do  work  at  constant  temperature,  energy  must  be  derived,  either 
from  the  system  itself,  or  else  from  its  surroundings.  This  energy  is 
known  as  available  energy.  The  maximum  external  work  done  by 
an  isothermal  process  is  generally  called  the  free  energy  of  the  pro- 
cess. It  is  very  important  to  distinguish  between  the  expressions, 
"  change  in  free  energy  "  and  "  change  in  total  or  internal  energy," 
since  it  is  only  under  special  conditions  that  the  two  expressions 
are  synonymous.  Thus,  certain  processes  are  known  in  which 
the  loss  in  free  energy  may  exceed  the  loss  in  total  energy,  and 


ELEMENTARY  PRINCIPLES  OF  THERMODYNAMICS     139 

other  processes  are  known  in  which  there  may  be  an  actual  gain 
in  total  energy  accompanying  a  loss  in  free  energy.  The  exact 
quantitative  relations  which  connect  the  change  in  free  energy 
with  the  corresponding  change  in  the  total,  or  internal,  energy 
associated  with  any  reversible  process,  is  given  by  the  Gibbs- 
Helmholtz  equation. 

If  we  combine  the  equation  formulating  the  first  law  of  ther- 
modynamics. 

U=Q-W, 

with  that  expressing  the  second  law, 


we  obtain 

(17) 


This  is  the  Gibbs-Helmholtz  equation.  It  is  one  of  the  most 
important  formulas  in  thermodynamics,  and  finds  numerous  ap- 
plications in  the  domain  of  physical  chemistry. 

In  certain  processes,  such,  for  example,  as  that  occurring  in  the 
Daniell  cell,  the  change  in  free  energy  with  the  temperature  is 

practically  zero:    i.e.,  [rr-m]  =  0>  and  consequently  U  =  —  W. 
\o  1  A 

In  other  words,  the  change  in  free  energy  and  the  change  in  total 
energy  are  numerically  equal,  when  the  temperature  coefficient 
of  free  energy  is  zero.  It  has  already  been  pointed  out,  that 
when  a  process  occurs  without  alteration  in  volume,  the  equa- 
tion expressing  the  first  law,  viz., 

U=Q-W, 
takes  the  form 

[/  =  «„ 

where  Qv  denotes  the  heat  absorbed  at  constant  volume.  If  this 
value  of  U  be  substituted  in  equation  (17),  we  obtain 


(is) 

which  is  a  useful  modification  of  the  Gibbs-Helmholtz  equation. 

Isothermal  and  Adiabatic  Processes.     When  the  temperature 

of  a  gas  is  maintained  constant,  by  supplying  heat  during  expan- 


140 


THEORETICAL  CHEMISTRY 


sion,  or  abstracting  heat  during  compression,  the  process  is  known 
as  an  isothermal  process.  On  the  other  hand,  when  a  gas  under- 
goes expansion  or  compression,  and  it  neither  receives  heat  from; 
nor  imparts  heat  to  its  surroundings,  the  process  is  known  as  an 

adiabatic  process.  If  a  gas  be  com- 
pressed adiabatically,  the  tempera- 
ture rises  and  tends  to  cause  the 
gas  to  expand;  this  temperature 
effect  opposes  the  compressing  force, 
and  therefore,  more  work  is  required 
to  bring  about  a  given  change  in 
volume  by  adiabatic  compression 
than  by  isothermal  compression. 
This  becomes  clear  if  we  make  use 
—i  of  the  diagram  shown  in  (Fig.  45) ,  in 
which  pressures  are  plotted  as  or- 
dinates,  and  volumes  as  abscissas. 


Volume 


Fig.  45 


If  a  gas  be  compressed  isothermally  from  volume  v\  to  volume  v2,  the 
work  done  will  be  represented  by  the  area  ABvzVi,  whereas  if  it  be 
compressed  adiabatically,  the  work  done  will  be  represented  by 
the  area  ACv2Vi.  It  will  be 
noticed  that  the  slope  of  the 
adiabatic  line,  AC,  is  steeper 
than  that  of  the  isothermal 
line,  AB. 

Garnet's  Cycle.  In  estab- 
lishing the  law  which  deter- 
mines the  maximum  quan- 
tity of  heat  which  can  be 
converted  into  work,  Carnot 
made  use  of  an  imaginary 
engine,  which  was  caused 
to  perform  a  reversible  cycle 
of  operations,  consisting  of 
two  isothermal  and  two 
adiabatic  volume  changes.  Such  a  cycle  of  operations  is  shown 
in  the  accompanying  diagram  (Fig.  46),  and  is  known  as  a  Carnot 
cycle.  The  four  successive  steps  involved  in  the  completion  of 
the  cycle  may  be  conveniently  considered  in  the  following 
order :  — 


G  H         F 
Fig.  46 


Volume 


ELEMENTARY  PRINCIPLES  OF  THERMODYNAMICS      141 

(1)  Let  A  represent  the  initial  state  of  the  system,  and  let  us 
suppose  that  it  expands  isothermally  at  the  temperature  T,  along 
AB,  through  an  infinitely  small  volume  do.     During  the  expan- 
sion, a  quantity  of  heat  Q,  will  be  withdrawn  from  the  surround- 
ings, and  a  quantity  of  work    represented  by  the  area  ABFG, 
will  be  done  by  the  system. 

(2)  The  system  is  next  allowed  to  expanb  adiabatically  along 
BC,  with  an  accompanying  fall  in  temperature  dT.     The  work 
done  in  this  stage  of  the   process  is  represented  by  the   area 
BCKF. 

(3)  The  system  is  now  compressed  isothermally  at  the  tem- 
perature T  —  dT,  along  CD,  with  an  evolution  of  a  quantity  of 
heat  Q',  slightly  less  than  Q.     The  work  done  upon  the  system 
in  this  stage  is  represented  by.  the  area  CDHK. 

(4)  Finally  the  system  is  compressed  adiabatically  along  DA, 
the  temperature  rising  until  the  initial  temperature  is  attained, 
and  the  system  is  restored  to  its  original  condition.     The  work 
done  on  the  system  during  the  last  stages  is  represented  by  the 
area  DAGH. 

The  net  work  done  by  the  system  is  represented  by  the  area 
A  BCD,  as  will  be  evident  from  the  diagram,  in  which  the  areas 
representing  work  done  by  the  system  are  shaded  from  left  to 
right,  while  the  areas  representing  work  done  upon  the  system  are 
shaded  from  right  to  left.  According  to  the  first  law  of  thermo- 
dynamics, the  net  work  done  in  the  cycle  must  be  equivalent 
to  the  heat  lost,  Q  -  Q'. 

Carnot's  cycle  is  the  only  completely  reversible  process  by  which 
an  indefinite  amount  of  work  can  be  performed  between  a  source 
and  a  condenser,  each  of  which  is  maintained  at  a  constant  tem- 
perature. 

The  ratio  of  the  quantity  of  work  produced  by  a  heat  engine, 
to  the  quantity  of  heat  absorbed  from  the  source,  is  known  as 
the  efficiency  of  the  engine.  Since  the  net  work  done  in  the  com- 
pletion of  a  Carnot  cycle  is  equivalent  to  Q  —  Q',  it  follows  that 
the  efficiency  of  an  engine  performing  such  a  cycle  will  be  given 

Q  —  Q' 
by  the  expression,  — -^ —  =  e. 

**? 

It  may  be  readily  shown,  that  of  all  heat  engines  working  be- 
tween two  given  temperatures,  that  which  is  perfectly  reversible 
has  the  maximum  efficiency  and,  furthermore,  that  all  reversible 


142  THEORETICAL  CHEMISTRY 

heat  engines  working  between  the  same  two  temperatures  have 
the  same  efficiency. 

The  Equation  of  Clapeyron.  It  may  be  proven  geometrically 
that  the  area  A  BCD  in  Fig.  46  is  equal  to  the  product  of  AE  and 
FG.  From  the  diagram  it  is  evident,  that  AE  represents  the 
increase  in  pressure  corresponding  to  an  increase  in  temperature 
dTy  when  the  system  is  maintained  at  constant  volume.  Ex- 
pressing this  mathematically,  we  have 

u-fi&a. 


In  like  manner,  FG  will  be  seen  to  represent  the  small  increase  in 
volume,  dv.  Therefore,  since  the  net  work  done  in  performing 
the  cycle  is  represented  by  the  area  A  BCD,  we  have 


dW  "\sr).dTJv-  (19) 

But,  as  has  already  been  proved  (see  equation  14), 


in  which  Q  denotes  the  heat  absorbed  in  a  reversible  cycle.  In 
the  cycle  we  are  now  considering,  the  quantity  of  heat  withdrawn 
from  the  source  may  be  expressed  by  the  partial  differential  coeffi- 


cient, dv:  therefore,  equation  (19)  becomes 


or 


The 


differential  coefficient,  ( — -  }   ,  may  be  interpreted  as  the  quan- 
\ovjT 

tity  of  heat  which  must  be  added  to  the  system  in  order  that  its 
temperature  may  remain  constant  during  unit  increase  in  volume. 
This  is  called  the  "latent  heat  of  expansion  ",  and  is  related  to 
the  term  "  latent  heat  "  as  usually  defined  in  the  following  man- 
ner: —  Let  Vi  and  vz  denote  the  volumes  of  one  gram  of  a  given 
substance  in  the  liquid  and  vapor  stages  respectively,  and  let 


ELEMENTARY  PRINCIPLES  OF  THERMODYNAMICS      143 

L  be  its  latent  heat,  i.e.,  the  quantity  of  heat  required  to  trans- 
form one  gram  of  the  substance  from  the  liquid  to  the  vapor  state, 
it  is  evident  that 


where  I  is  the  latent  heat  of  expansion.     Therefore,  equation  (20) 
becomes, 


5TJV  •    TbvT      T      Tfo-vJ 

Since  the  pressure  of  a  saturated  vapor  is  independent-  of  its  vol- 
ume, provided  the  liquid  phase  is  present,  the  partial  differential 
coefficient  may  be  replaced  by  the  total  differential  coefficient, 
and  equation  (21)  becomes, 


dT 


(22) 


This  is  the  well-known  equation  of  Clapeyron.  It  is  of  value 
in  the  application  of  thermodynamics  to  processes  involving  a 
change  of  state.  For  example;  it  enables  one  to  calculate  the 
heat  of  vaporization  of  a  liquid,  provided  the  specific  volumes  of 
the  liquid  and  vapor  phases,  together  with  the  rate  of  change  of 
the  vapor  pressure  with  the  temperature,  are  known. 

As  an  illustration  of  the  use  of  equation  (22),  let  us  calculate  the 
heat  of  vaporization  of  water  at  its  boiling-point  from  the  following 
data:  —  The  vapor  pressure  of  water  is  746.52  mm.  at  99.5°  and 
773.68  mm.  at  100.5°;  the  specific  volume  of  water  vapor  at  100° 
is  1650.8  cc.,  and  the  specific  volume  of  liquid  water  at  100°  is  1.04 
cc.  Therefore,  the  value  of  dp/dT  at  100°  will  be,  773.68  - 
746.52  =  2.716  cm.,  or  2.716  X  13.6  X  981  =  36,239  dynes/cm.2. 
Substituting  the  above  values  in  equation  (22),  we  have 

L  =  36,239  X  373  (1,650.8  -  1.04) 
=  22.3  X  109  ergs. 

On  dividing  this  quantity  by  the  mechanical  equivalent  of  heat, 
J  =  4.183  X  107  ergs,  we  obtain  L  =  533.1  calories  per  gram, 
as  the  heat  of  vaporization  of  water. 

In  like  manner,  the  equation  may  also  be  applied  to  the  process 
of  fusion.  For  example,  let  us  calculate  the  change  in  the  freez- 


144  THEORETICAL  CHEMISTRY 

ing-point  of  water  due  to  an  increase  in  pressure  of  one  atmosphere. 
The  specific  volume  of  water  at  0°  is  1.000  cc.,  and  the  specific  vol- 
ume of  ice  at  the  same  temperature  is  1.092  cc.:  hence,  v2  —  v\  = 
1.000  -  1.092  =  -  0.092  cc.  The  heat  of  fusion  of  ice  is  79.6 
calories  per  gram,  or  79.6  X  4.183  X  107  ergs  per  gram.  Sub- 
stituting in  equation  (22),  we  have 

dp       79.6  X  4.183  X  107 
dT~  273  X  (-  0.092) 

=  -  1.33  X  108  dynes/cm.2 

That  is,  1.33  X  108  dynes/cm.2,  is  the  pressure  required  to  lower 
the  freezing-point  of  water  1°;  therefore,  its  reciprocal, 


will  be  the  lowering  in  the  freezing-point  in  degrees  produced  by 
an  increase  in  pressure  of  one  dyne,  and,  since  one  atmosphere  is 
equivalent  approximately  to  1  X  106  dynes,  it  follows  that  the 
lowering  in  the  freezing-point  of  water  produced  by  an  increase  in 
pressure  of  one  atmosphere  will  be,  0.75  X  10~8  X  106  =  0.0075°. 
The  negative  sign  indicates  that  increase  of  pressure  lowers  the 
freezing-point  ;  this  is  due  to  the  fact,  that  the  specific  volume  of 
ice  is  greater  than  the  specific  volume  of  water  at  the  freezing 
temperature. 

The  Equation  of  Clausius-Clapeyron.  If  we  assume  that  the  gas 
laws  apply  to  vapors,  as  we  may  under  ordinary  conditions,  and  if 
we  regard  the  specific  volume  of  a  substance  in  the  liquid  state  as 
negligible  in  comparison  with  the  specific  volume  of  its  vapor, 
we  may  substitute,  RT/p,  for  v2  in  the  equation  of  Clapeyron, 
and  obtain  the  equation, 


dT      RT* 

On  rearrangement  this  becomes, 

J_    <!v=     X 
dT'  p  ~  RT2 

or 


ELEMENTARY  PRINCIPLES  OF  THERMODYNAMICS      145 

This  is  known  as  the  equation  of  Clausius-Clapeyron,  and,  like  the 
equation  from  which  it  is  derived,  it  finds  numerous  applications 
in  physical  chemistry. 

Entropy.     The  maximum  amount  of  work  obtainable  from  an 
ideal  heat  engine  operating  in  a  reversible  cycle  between  the  tem- 

f)  (  rrt rr\  \ 

peratures  TI  and  T2)  has  been  shown  to  be     ^    * ».  where 

1 1 

Q  denotes  the  quantity  of  heat  delivered  to  the  engine.  Expand- 
ing this  expression,  we  have 

c-^ -«(£)-«©=«-(?>• 

In  this  equation,  the  term,  (TFT)  Tz,  represents  the  quantity  of  heat 

which  is  not  convertible  into  work,  or,  in  other  words,  it  repre- 
sents the  unavailable  energy.  It  is  obvious  that  the  unavail- 
able energy  will  become  zero  only  when  T%  =  0.  Hence,  if  the 
temperature  of  the  condenser  could  be  reduced  to  the  absolute 
zero,  the  available  energy  would  be  represented  by  Q,  and  the 
efficiency  of  the  engine  would  become  100  per  cent. 

The  quantity-^-  =  S,  is  called  the  entropy  of  the  system  and, 
1 1 

like  the  quantities,  U  and  W,  its  value  is  wholly  dependent  upon 
the  state  of  the  system,  and  not  upon  the  path  by  which  that 
state  is  reached.  We  may  define  entropy  as  that  quantity  which 
multiplied  by  the  lowest  available  temperature  gives  the  unavail- 
able energy.  Although  the  total  entropy  of  a  system  cannot  be 
measured,  it  is  possible  to  determine  the  difference  in  entropy  be- 
tween two  given  states.  Therefore,  entropy  is  always  expressed 
as  an  increase,  or  a  decrease,  above  or  below  some  arbitrarily 
chosen  standard.  Since  during  an  adiabatic  change,  no  heat 
enters  or  leaves  a  system,  it  follows  that  dQ  =  0,  or  dQ/T  =  0, 
and  therefore,  Q/T  is  constant.  Hence,  an  adiabatic  may  be 
regarded  as  a  line  of  constant  entropy.  It  is  on  account  of  this 
property,  that  adiabatic  lines  are  frequently  called  isentropic  lines. 
Since  adiabatic  lines  are  lines  of  constant  entropy,  it  is  cus- 
tomary in  measuring  the  entropy  of  a  system,  to  select  some 
one  adiabatic,  as  a  reference  line,  to  which  all  other  adiabatics 
of  the  system  may  be  referred.  The  pressure- volume  diagram 
of  a  given  mass  of  a  substance  may  be  ruled  off  into  a  series  of 


146  THEORETICAL  CHEMISTRY 

adiabatic  lines,  representing  successive  states  of  the  substance 
which  differ  by  definite  units  of  entropy,  in  just  the  same 
manner  as  the  states  represented  by  the  isothermal  lines  differ 
by  definite  units  of  temperature. 

Since,  in  every  physical  or  chemical  process  occurring  in  nature, 
transference  of  heat  is  accompanied  by  a  fall  in  temperature, 
it  follows  that,  Q/T,  must  undergo  a  corresponding  increase.  In 
other  words,  in  an  irreversible  process,  the  entropy  tends  toward 
a  maximum.  When  this  maximum  is  reached,  the  available 
energy  will  be  zero,  and  the  temperature  of  the  component  parts 
of  the  system  will  all  be  at  the  same  temperature. 

In  an  'isothermal  reversible  process,  -jjj-  will  have  a  finite  value 

and  the  increase  in  entropy,  in  passing  from  a  state  A  to  a  state 
Bj  will  be  represented  by  the  equation, 

BdQ      Q 


but 

j,  =  j-fj?  ,  (See  equation  14) 

therefore, 


That  is,  the  increase  in  the  entropy  of  the  system  is  equivalent  to 
the  rate  at  which  the  free  energy  decreases  with  the  temperature. 

REFERENCES 
A  System  of  Physical  Chemistry.     W.  C.  McC.  Lewis.     Vol.  II  Chapters 

I  and  II. 

Introduction  to  General  Thermodynamics.     Perkins. 
Thermochemistry  and  Thermodynamics.     Sackur.     Translated  by  Gibson. 

Chapters  III,  IV  and  V. 
Thermodynamics  and   Chemistry.     Macdougall.     Chapters   I    to   VIII   in- 

clusive. 

Theory  of  Thermodynamics.     Buckingham.     Chapters  I  to  VIII  inclusive. 
A  Text-book  of  Thermodynamics.     Partington.     Chapters  I,  II  and  III. 
Chemical  Thermodynamics.     G.  N.  Lewis. 

PROBLEMS 

1.  A  piston  whose  diameter  is  20  cm.  moves  10  cm.  under  a  pressure 
of  10  atmospheres.  How  many  calories  of  heat  are  transformed  into 
work? 


ELEMENTARY  PRINCIPLES  OF  THERMODYNAMICS     147 

2.  In  a  certain  thermo dynamic  process  the  system  gains  600  calories 
of  internal  energy,  and  at  the  same  time  1200  X  107  e^p  of  work  are 
done;  how  much  heat  must  be  supplied?    What  is  the  efficiency  of  the 
process? 

3.  How  many  ergs  are  equivalent  to  5  calories? 

4.  A  Carnot  engine  takes  in  100  calories  and  gives  out  80  calories. 
How  many  ergs  of  work  are  done,  and  what  is  the  efficiency  of  the  engine? 

5.  A  Carnot  engine,  when  reversed,  delivers  150  calories  to  the  source. 
If  300  X  107  ergs  are  expended,  calculate  the  amount  of  heat  taken  from 
the  condenser. 

6.  How  much  work,  expressed  in  ergs,  can  an  ideal  engine  perform, 
working  between  the  temperatures  of  150°  and  20°,  provided  180  calories 
are  delivered  to  the  engine  at  the  higher  temperature?    Also  calculate 
the  efficiency  of  the  operation. 

\x7.  Jhe  density  of  solid  phenol  is  1.072,  and  the  density  of  liquid  phenol 
isfl.0561;   its  heat  of  fusion  is  24.93  calories  per  gram,  and  its  freezing- 
int  is  34°.     Find  the  change  in  the  freezing-point  produced  by  an  in- 
crease in  pressure  of  1  atmosphere. 

8.  Acetic  acid  melts  at  16.6°,  dT/dp  =  0.0242°  per  atmosphere,  and 
the  heat  of  fusion  is  46.42  calories  per  gram.     Calculate  the  change  in 
volume  accompanying  the  liquefaction  of  1  gram  of  acid. 

9.  From  the  following  data  calculate  the  value  of  dpi  AT  for  ethyl 
propionate:  —  density  in  liquid  state,  0.7958,  density  in  vapor   state, 
0.0033,  both  of  the  foregoing  values  corresponding  to  the  boiling-point 
of  ethyl  propionate,  99°;  the  heat  of  vaporization  is  77  calories  per  gram. 

10.  The  heat  of  vaporization  of  ether  is  88.39  calories  per  gram  at 
34.5°.     Calculate  the  change  in  boiling-point  due  to  a  change  in  pressure 
from  760  mm.  to  750  mm. 

(NOTE.  In  equation  (23)  let  R  =  2  calories,  and  replace  L,  the  heat 
of  vaporization  per  gram,  by  X  =  LM ,  the  heat  of  vaporization  per  gram- 
molecular  weight,  M.) 

11.  Compute  the  heat  of  vaporization  of  ethyl  formate  from  the  follow- 
ing data:  —  dp/dT  =  27  mm.,  the  boiling-point  is  54.3°  at  760  mm.,  and 
the  molecular  weight  is  74. 

12.  The  heat  of  fusion  of  ice  is  80  calories  per  gram,  and  the  heat  of 
vaporization  of  water  is  597  calories  per  gram  at  the  same  temperature, 
viz.,  0°.    What  is  the  heat  of  sublimation  of  ice  at  0°? 

(NOTE.  It  follows  from  the  first  law  of  thermodynamics  that  the  heat 
of  sublimation  at  the  freezing-point  must  be  equal  to  the  sum  of  the  heats 
of  fusion  and  vaporization,  i.e.,  it  is  immaterial,  so  far  as  the  expenditure 
of  energy  is  concerned,  whether  a  substance  passes  directly  from  the  solid 
to  the  gaseous  state,  or  whether  it  passes  first  into  the  liquid  state  and 
then  into  the  gaseous  state.) 


148  THEORETICAL  CHEMISTRY 

13.  The  vapor  pressure  of  both  ice  and  water  at  0°  is  4.62  mm.,  and 
the  heat  of  fusion  of  ice  is  80  calories  per  gram.     Calculate  the  difference 
between  the  temperature  coefficients  of  vapor  pressure  for  the  two  states. 

14.  Calculate  the  heat  of  sublimation  of  ice  from  the  data  of  the  pre- 
ceding problem,  together  with  the  following  data:—-  The  specific  vol- 
umes of  saturated  water  vapor  and  ice  are  210,680  cc.  per  gram  and  1.092 
cc.  per  gram  respectively  at  the  vapor  pressure  4.619  mm.,  and  the  change 
in  vapor  pressure  per  degree  is  0.3852  mm. 

15.  Calculate  the  lowering  of  the  freezing-point  of  ice  when  it  is  sub- 
jected to  a  pressure  of  60  atmospheres.     Make  use  of  data  given  in  pre- 
ceding problems. 

16.  Calculate  the  change  in  free  energy  when  one  gram-molecule  of 
a  liquid  is  evaporated  at  constant  temperature  T,  and  constant  pressure 
p,  if  Vi  and  t>2  are  the  volumes  of  one  gram-molecule  of  the  liquid  and 
vapor    respectively.     For    water,   when    T  =  373,    p  —  1    atmosphere, 
Vi  =  18.02  cc.  and  y2  =  18.02  X  1674  cc.     Express  your  answer  in  cal- 
ories. 

17.  A  substance  changes  state  at  80°  by  the  addition  of  500  calories. 
What  is  the  gain  in  entropy? 

18.  A  system  at  80°  receives  100  calories.     At  60°  it  rejects  75  calo- 
ries.    What  has  been  the  gain  in  entropy? 

19.  Calculate  the  change  in  free  energy  of  a  system  under  atmospheric 
pressure  and  at  20°,  when  its  internal  energy  increases  by  500  calories;  its 
volume  changes  from  2  to  5  liters,  and  its  entropy  gains  2  units. 

20.  Calculate  the  change  in  entropy  when  10  grams  of  mercury  solidify, 
having  given  the  freezing-point  of  mercury,  —  38.3°,  and  its  heat  of  fusion, 
2.82  calories  per  gram.     Does  it  gain  or  lose  entropy  in  the  process? 


CHAPTER  VII 
SOLUTIONS 

Definition  of  Terms.  Having  dealt  with  the  properties  of 
pure  substances  in  the  gaseous,  liquid  and  solid  states  we  now 
proceed  to  the  consideration  of  the  properties  of  mixtures  of  two 
or  more  pure  substances.  When  such  a  mixture  is  chemically 
and  physically  homogeneous,  and  no  abrupt  change  in  its  prop- 
erties results  from  an  alteration  of  the  proportions  of  the  com- 
ponents of  the  mixture,  it  is  termed  a  solution. 

A  solution  may  also  be  defined  as  a  homogeneous  phase  composed 
of  two  or  more  molecular  species.  Thus,  when  alcohol  is  added 
to  water,  we  obtain,  when  equilibrium  has  been  established,  a 
homogeneous,  single-phase  system  composed  of  two  different 
molecular  species,  uniformly  and  intim^ely  mixed  with  each 
other.  Such  a  system  constitutes  a  <xtrue  "  solution  as  distin- 
guished from  so-called  "  colloidal  "  solutions  which  are  to  be  con- 
sidered as  occupying  an  intermediate  position  between  true 
solutions,  on  the  one  hand,  and  heterogeneous  mixtures,  on  the 
other.  The  subject  of  colloidal  solutions  will  be  discussed  in  a 
subsequent  chapter. 

When  one  substance  is  dissolved,  or  molecularly  dispersed, 
in  another,  it  is  customary  to  designate  as  the  solvent  that  com- 
ponent which  is  present  in  the  larger  proportion,  and  to  call  the 
other  component  the  solute.  It  should  be  remembered  that  this 
method  of  designation  is  purely  arbitrary,  and  that  the  ability 
of  one  substance  to  dissolve  in  another  does  not  imply  that  one 
component  possesses  greater  solvent  power  than  .the  other.  In 
fact,  it  frequently  happens  that  two  liquids  are  miscible  in  all 
proportions,  in  which  case  either  one  may  be  regarded  as  the  sol- 
vent. Such  liquids  are  said  to  be  consolute. 

While  no  general  principle  has  as  yet  been  discovered  by  means 
of  which  the  solubility  of  one  substance  in  another  can  be  pre- 
dicted, it  is  nevertheless  true  that  solubility  is  to  a  large  extent 
dependent  upon  the  existence  of  a  similarity  in  character  between 

149 


150  THEORETICAL  CHEMISTRY 

the  components  of  a  solution.  For  example,  benzene  is  insoluble 
in  water  but  mixes  in  all  proportions  with  its  homologue,  toluene. 
Likewise,  a  great  number  of  organic  compounds,  such  as  the 
solid  hydrocarbons  and  fats,  which  are  insoluble  in  water,  dissolve 
readily  in  ether  or  benzene.  On  the  other  hand,  numerous  in- 
organic salts  dissolve  in  water  over  a  wide  range  of  concentration, 
notwithstanding  the  fact  that  they  bear  little  or  no  resemblance 
to  the  solvent 

Modes  of  Expressing  the  Composition  of  Solutions.  There  are 
several  different  ways  in  which  the  composition  of  a  solution  may 
be  expressed.  A  very  common  mode  of  expressing  the  composi- 
tion of  a  dilute  solution  is  in  terms  of  gram-molecular  weight, 
formula-weight  or  equivalent-weight  per  liter  of  solution. 

Thus,  the  concentration  of  a  solution  of  sulphuric  acid,  (H2S04 
=  2  X  1.008  +  32.06  +  4  X  16  =  98.08),  containing  98.08  grams 
per  liter  is  said  to  be  molar  or  formal,  whereas  the  concentration 
of  a  solution  which  contains  one  equivalent,  or  49.04  grams,  of 
acid  per  liter  is  said  to  be  equivalent  or  normal.  It  is  sometimes 
convenient  to  express  the  concentration  of  a  solution  in  terms  of 
1000  grams  of  solvent  rather  than  in  terms  of  1000  cc.  of  solution. 
When  concentrations  are  expressed  on  the  basis  of  the  weight 
of  the  solvent,  they  are  known  as  weight  concentrations  to  dis- 
tinguish them  from  the  volume  concentrations  previously  de- 
fined. It  is  obvious  that  volume  concentrations  can  be  converted 
into  weight  concentrations  provided  the  densities  of  the  solutions 
are  known. 

The  composition  of  more  concentrated  solutions  may  be  ex- 
pressed, either  in  percentages,  or  in  terms  of  mol  fractions,  pref- 
erably the  latter.  By  the  mol  fraction  of  a  dissolved  substance 
is  meant  the  number  of  mols  of  the  substance  divided  by  the 
total  number  of  mols  in  the  solution^  Thus,  if  a  solution  is  com- 
posed  of  nA  mols  of  A  and  nB  .mols  of  B,  the  mol  fractions  of  the 
two  components  will  be  as  follows :  — 

XA  = —  =  mol  fraction  of  A, 

nA  +  nB 

and 

XB  =  -          -  =  mol  fraction  of  B. 

From  these  two  expressions  it  is  evident  that  XA  -f  xs  =  1. 


SOLUTIONS  151 

Classification  of  Solutions.     There  are  nine  possible  classes  of 
solutions,  as  follows :  — 

(1)  Solution  of  gas  in  gas; 

(2)  Solution  of  liquid  in  gas; 

(3)  Solution  of  solid  in  gas; 

(4)  Solution  of  gas  in  liquid; 

(5)  Solut'on  of  liquid  in  liquid; 

(6)  Solution  of  solid  in  liquid; 

(7)  Solution  of  gas  in  solid; 

(8)  Solution  of  liquid  in  solid;    . 

(9)  Solution  of  solid  in  solid. 

While  examples  of  all  of  these  different  types  of  solutions  are 
known,  only  the  more  important  classes  will  be  considered  here. 
Solutions  of  Gases  in  Gases.  In  solutions  of  this  class,  the 
components  may  be  present  in  any  proportions,  since  gases  are 
completely  miscible.  In  a  mixture  of  gases  where  no  chemical 
action  occurs,  each  gas  behaves  independently,  the  properties  of 
the  gaseous  mixture  being  the  sum  of  the  properties  of  the  con- 
stituents. Thus,  the  total  pressure  of  a  mixture  of  several  gases  is 
equal  to  the  sum  of  the  pressures  which  eacH  gas  wouTd  exerTwere  it 
alone  present  in  the  volume  occupied  by  the  mixture.  This  law  was 
discovered  by  Dalton*  and  is  known  as  Dalton's  law  of  partial 
pressures.  If  the  partial  pressures  of  the  constituent  gases  be 
denoted  by  pit  p2,  ps,  etc.,  and  P  and  V  represent  the  total  pres- 
sure and  the  total  volume  of  the  gaseous  mixture,  then 

PV  =  V  (Pl  +  p2  +  PS  +  • .  •  •)•  (1) 

Dalton's  law  holds  when  the  partial  pressures  are  not  too  great, 
its  order  of  validity  being  the  same  as  that  of  the  other  gas  laws. 
Dalton's  law  can  be  tested  experimentally  by  comparing  the 
total  pressure  of  the  gases  with  the  sum  of  the  pressures  exerted 
by  each  gas  before  mixture.  The  possibility  of  measuring  the 
partial  pressure  of  one  of  the  two  components  of  a  gas  mixture, 
provided  a  diaphragm  could  be  found  which  would  be  pervious 
to  one  of  the  gases  but  not  to  the  other,  was  first  pointed  out  by 
van't  Hoff.  It  was  shown  by  Ramsay,  f  that  the  walls  of  a 

*  Gilb.  Ann.,  12,  385  (1802). 
f  Phil.  Mag.  (5),  38,  206  (1894). 


152 


THEORETICAL  CHEMISTRY 


vessel  of  palladium,  when  sufficiently  heated,  permit  the  free 
passage  of  hydrogen  but  not  of  nitrogen.  The  walls  are  said  to 
be  semi-permeable.  A  sketch  of  the  apparatus  used  by  Ramsay 
in  the  verification  of  Dalton's  law  is  shown  in  Fig.  47.  A  small 
vessel  of  palladium,  P,  containing  nitrogen,  is  connected  with 
a  manometer  AB,  which  serves  to  measure  the  pressure  of  the 
gas  in  P.  The  vessel  P,  is  enclosed  within  a  larger  vessel  C, 
which  can  be  filled  with  hydrogen  at  known  pressure.  On  heating 
P  and  passing  a  current  of  hydrogen  at  a  definite  pressure  through 
C,  the  hydrogen  enters  P  until  the  pressures  due  to  hydrogen  in- 
side and  outside  are  equal.  The 
total  pressure  in  P,  measured  on 
the  manometer,  is  greater  than  the 
pressure  in  C.  The  difference 
between  the  two  pressures  is  very 
nearly  equal  to  the  partial  pressure 
of  the  nitrogen.  Conversely,  if  a 
mixture  of  the  two  gases  be  intro- 
duced into  P,  which  is  then  heated 
and  maintained  at  sufficiently  high 
temperature  to  insure  its  perme- 
ability to  hydrogen,  the  partial 
pressure  of  the  nitrogen  can  be 
determined  by  passing  a  current 
of  hydrogen  at  known  pressure 
through  C  until  equilibrium  is  at- 
tained, as  shown  by  the  manom- 
eter. The  difference  between  the 


Hydrogen 


Fig.  47 


external  and  internal  pressures  is  the  partial  pressure  of  the 
nitrogen.  This  experiment  has  a  very  important  bearing  upon 
the  modern  theory  of  solution. 

Solutions  of  Gases  in  Liquids.  The  solubility  of  gases  in 
liquids  is  limited,  the  extent  to  which  they  dissolve  depending 
upon  the  pressure,  the  temperature,  the  nature  of  the  gas,  and 
the  nature  of  the  solvent.  When  a  liquid  cannot  absorb  any 
more  of  a  gas  at  a  definite  temperature,  it  is  said  to  be  saturated, 
and  the  solution  is  called  a  saturated  solution.  The  solubility  of 
a  gas  in  a  liquid  is  defined  by  Ostwald  as  the  ratio  of  the  volume 
of  the  gas  absorbed  to  the  volume  of  the  absorbing  liquid  at  a 
specified  temperature  and  pressure.  If  the  solubility  of  the  gas 


SOLUTIONS 


153 


be  represented  by  S,  we  have 


a  =v/v,  (2) 

where  v  is  the  volume  of  gas  absorbed  and  V  is  the  volume  of  the 
absorbing  liquid.  The  "  absorption  coefficient  "  of  Bunsen,  in 
terms  of  which  he  expressed  the  results  of  his  measurements  of 
the  solubility  of  gases,  may  be  defined  as  the  volume  of  a  gas, 
reduced  to  0°  C.  and  76  cm.  pressure  which  is  absorbed  by  unit 
volume  of  a  liquid  at  a  given  temperature  and  under  a  pressure 
of  76  cm.  of  mercury.  In  certain  cases  the  volume  of  the  gas 
absorbed  is  found  to  be  independent  of  the  pressure,  so  that  if  a 
is  the  coefficient  of  gaseous  expansion,  and  /?,  Bunsen's  coefficient 
of  absorption,  then 

8  =  0  (1  +  cA).  (3) 

The  solubilities  of  a  few  gases  in  water  and  alcohol  as  determined 
byJBunsen  are  given  in  the  following  table:  — 


SOLUBILITY  OF  GASES 


Gas. 

Water. 

Alcohol. 

0° 

15° 

0° 

15° 

Hydrogen 

0.0215 
0.0489 
1.797 

0.0190 
0.0342 
0.1002 

0.0693 
0.2337 
4.330 

0.0673 
0.2232 
3.199 

Oxy  ge  n 

Carbon  dioxide   .  . 

The  solubility  of  gases  in  water  is  appreciably  diminished  by 
the  presence  of  dissolved  solids  or  liquids,  especially  electrolytes. 
Various  theories  have  been  proposed  to  account  for  the  diminished 
solubility  of  gases  in  salt  solutions  but  the  most  satisfactory  is 
that  due  to  Philip,*  who  suggests  that  the  phenomenon  is  caused 
by  the  hydration  of  the  dissolved  salt.  A  portion  of  the  water 
in  the  salt  solution  is  supposed  to  be  in  combination  with  the  salt, 
the  water  which  is  thus  removed  from  the  role  of  solvent,  being 
no  longer  free  to  absorb  gas. 

The  solubility  of  a  gas  increases  with  increase  in  pressure. 
For  gases  which  do  not  react  chemically  with  the  solvent, 
there  exists  a  simple  relation  between  pressure  and  solubility, 

*  Trans.  Faraday  Soc.,  3,  140  (1907). 


154  THEORETICAL  CHEMISTRY 

discovered  by  Henry.*  This  relation,  known  as  Henry's  Law 
may  be  stated  as  follows :  —  When  a  gas  is  absorbed  in  a  liquid, 
the  weight  dissolved  is  proportional  to  the  pressure  of  the  gas.  Since 
pressure  and  volume,  at  constant  temperature,  are  inversely 
proportional  (Boyle's  law),  the  law  of  Henry  may  be  stated 
thus :  —  The  volume  of  a  gas  absorbed  by  a  given  volume  of 
liquid  is  independent  of  the  pressure.  There  is  yet  another 
form  in  which  the  law  may  be  stated  which  is  instructive  in 
connection  with  the  modern  theory  of  solution.  When  a  definite 
volume  of  liquid  is  saturated  with  a  gas  at  constant  temperature 
and  pressure,  a  condition  of  equilibrium  is  established  between  the 
gas  in  solution  and  that  in  the  free  space  over  the  solution,  there- 
fore, Henry's  law  may  be  stated  as  follows :  —  The  concentration 
of  the  dissolved  gas  is  directly  proportional  to  that  in  the  free  space 
above  the  liquid.  If  Ci  represents  the  concentration  of  the  gas  in 
the  liquid  and  c?  the  concentration  in  the  free  space  above  the 
liquid,  Henry's  law  may  be  expressed  thus, 

Ci/c2  =  k,  (4) 

where  k  is  known  as  the  solubility  coefficient. 

Dalton  showed  that  the  solubility  of  the  individual  gases  in  a 
mixture  of  gases  is  directly  proportional  to  their  partial  pressures, 
the  solubility  of  each  gas  being  nearly  independent  of  the  pres- 
ence of  the  others. 

As  will  be  seen  from  the  foregoing  table,  the  solubility  of  a  gas 
in  a  liquid  diminishes  with  increase  in  temperature,  f  Concerning 
the  influence  of  the  nature  of  the  gas  on  its  solubility,  it  may  be 
said  that  those  gases  which  exhibit  acid  or  basic  reactions  are 
the  most  soluble,  the  solubilities  of  neutral  gases  being  small. 
In  the  case  of  many  of  the  very  soluble  gases  Henry's  law  does 
not  hold.  For  example,  ammonia,  a  gas  having  marked  basic 
properties  and  a  large  coefficient  of  solubility,  does  not  obey 
Henry's  law  at  ordinary  temperatures,  the  mass  of  ammonia 
absorbed  not  being  proportional  to  the  pressure.  The  curve 
showing  the  variation  in  solubility  with  pressure  at  0°  C.  has 
two  marked  discontinuities.  At  temperatures  above  100°  C. 

*  Gilb.  Ann.,  20,  147  (1805). 

t  It  has  been  claimed  that  helium  is  an  exception  to  the  rule  that  the 
solubility  of  a  gas  in  a  liquid  diminishes  with  increase  in  temperature.  The 
absorption  coefficient  of  He  diminishes  from  0°  to  25°  and  then  increases 
again  as  the  temperature  is  raised. 


SOLUTIONS  155 

the  gas  obeys  Henry's  law.     Sulphur  dioxide  behaves  similarly, 
the  law  holding  only  for  temperatures  exceeding  40°  C. 

With  regard  to  the  connection  between  the  solvent  power  of  a 
liquid  and  its  nature,  but  little  is  known.  About  all  that  can  be 
said  is,  that  the  order  of  solubility  of  gases  in  different  liquids  is 
almost  always  the  same.  Thus  in  the  preceding  table  the  solu- 
bilities of  hydrogen,  oxygen  and  carbon  dioxide  in  water  and  in 
alcohol  will  be  seen  to  be  approximately  proportional.  A  slight 
change  in  volume  always  results  when  a  gas  is  dissolved  in  a 
liquid.  In  general,  it  may  be  said  that  the  less  compressible  a 
gas  is,  the  greater  is  the  increase  in  volume  produced  when  it 
is  absorbed  by  a  liquid.  It  is  of  interest  to  note  that  the  increase 
in  volume  caused  by  the  solution  of  a  gas  is  nearly  equal  to  the 
corresponding  value  of  6  in  the  equation  of  van  der  Waals.  This 
is  shown  in  the  following  table:  — 

INCREASE  IN  VOLUME  DUE  TO  ABSORPTION  OF  GAS 


Gas. 

Increase  in  Vol. 

b 

Oxvsren 

0  00115 

0  000890 

Nitrogen                       

0  00145 

0  001359 

Hydrogen                   

0  00106 

0  000887 

C&rbon  dioxide 

0  00125 

0  000866 

Solutions  of  Liquids  in  Liquids.  Solutions  of  liquids  in  liquids 
can  be  divided  into  three  classes  as  follows:  —  (1)  Liquids  which 
are  miscible  in  all  proportions;  (2)  Liquids  which  are  partially 
miscible;  and  (3)  Liquids  which  are  immiscible.  Examples  of 
these  three  classes  in  the  order  mentioned  are,  alcohol  and  water, 
ether  and  water,  and  benzene  and  water.  As  to  the  cause  of  the 
miscibility  and  non-miscibility  of  liquids  very  little  is  known. 

If  a  small  amount  of  ether  is  added  to  a  large  volume  of  water 
in  a  separatory  funnel  and  the  mixture  vigorously  shaken,  a 
perfectly  homogeneous  solution  will  be  obtained.  On  gradually 
increasing  the  amount  of  ether,  shaking  after  each  addition,  a 
concentration  will  eventually  be  reached  at  which  a  separation 
into  two  layers  will  take  place.  The  upper  layer  is  a  saturated 
solution  of  water  in  ether,  and  the  lower  layer  is  a  saturated  solu- 
tion of  ether  in  water.  So  long  as  the  relative  amounts  of  the 
two  liquids  is  such  that  the  mixture  does  not  become  homogeneous 


156 


THEORETICAL  CHEMISTRY 


on  standing,  the  composition  of  the  two  layers  will  be  independent 
of  the  relative  amounts  of  the  two  components.  Measurements 
of  the  mutual  solubility  of  liquids  have  been  made  by  Alexieeff* 
by  placing  weighed  amounts  of  two  different  liquids  in  sealed 
tubes  and  observing  the  temperature  at  which  the  mixture  be- 
came homogeneous.  In  general  the  solubility  of  a  pair  of  par- 
tially miscible  liquids  increases  with  the  temperature,  and  there- 
fore it  may  be  inferred  that  at  a  sufficiently  high  temperature 


100* 


Percentage  Water  in  Phenol 


Percentage  Phenol  in  Water 

Fig.  48 


100* 


the  mixture  will  become  perfectly  homogeneous.  An  example 
of  this  type  of  binary  mixture  is  furnished  by  phenol  and 
water,  the  solubility  curve  of  which  is  shown  in  Fig.  48.  In 
this  diagram  temperature  is  plotted  on  the  axis  of  ordinates 
and  percentage  composition  of  the  solution  on  the  axis  of  ab- 
scissae. Starting  with  a  small  amount  of  phenol  and  adding  it 
in  increasing  quantities  to  a  large  volume  of  water,  a  concen- 
tration will  eventually  be  reached  at  which  the  solution  will 
separate  into  two  layers.  This  concentration  is  represented  by 
the  point  A.  On  raising  the  temperature,  the  solubility  of  phenol 
in  water  increases,  as  shown  by  the  curve  AB.  In  like  manner, 
starting  with  pure  phenol  and  adding  increasing  amounts  of  water, 
separation  into  two  layers  will  occur  at  a  concentration  repre- 

*  Jour,  prakt.  Chem.,  133,  518  (1882);  Bull.  Soc.  Chem.,  38,  145  (1882). 


SOLUTIONS  157 

sented  by  the  point  C.  As  the  temperature  is  raised,  the  solu- 
bility of  water  in  phenol  increases,  as  shown  by  the  curve  CB. 
When  the  temperature  is  raised  above  68°. 4  C.,  corresponding  to 
the  point  B,  phenol  and  water  become  miscible  in  all  proportions. 

If  we  start  with  a  solution  whose  temperature  and  composition 
is  represented  by  the  point  a,  the  addition  «of  increasing  amounts 
of  phenol,  at  constant  temperature  will  be  represented  by  the 
dotted  line  afed.  When  the  point  /  is  reached,  the  solution  will 
separate  into  two  layers  the  composition  of  which  will  be  inde- 
pendent of  the  relative  amounts  of  phenol  and  water.  At  e  the 
solution  will  again  become  homogeneous.  If  the  solution  repre- 
sented by  the  point  a  be  again  chosen  as  the  starting  point,  and  its 
composition  be  kept  unaltered  while  the  temperature  is  raised  to  a 
value  above  68°.4  C.,  the  change  will  be  represented  by  the  dotted 
line  ab.  If  now  the  temperature  be  maintained  constant  and  the 
percentage  of  phenol  increased,  the  alteration  in  composition  will 
be  effected  without  discontinuity,  as  represented  by  the  dotted 
line  be.  On  cooling  the  solution  represented  by  the  point  c  to  the 
initial  temperature  of  a,  the  point  d  will  be  reached.  Thus  it  is 
possible  to  pass  from  a  to  d  by  the  path  abed  without  causing  a 
separation  of  the  components  into  two  layers.  There  is  an 
analogy  between  the  solubility  curve  of  a  pair  of  partially  mis- 
cible liquids  and  the  dotted,  parabolic  curve  in  the  diagram  of 
the  isothermals  of  carbon  dioxide,  shown  in  Fig.  11.  In  both 
cases  there  is  but  one  phase  outside  of  the  curves,  while  two  phases 
are  coexistent  within  the  area  enclosed  by  the  curves.  The 
analogy  may  be  traced  further,  since  in  each  case  only  one  phase 
can  exist  above  a  certain  temperature.  The  temperature  corre- 
sponding to  the  apex  of  the  parabolic  curve  in  Fig.  11,  is  termed 
the  critical  temperature  of  carbon  dioxide  and  by  analogy  the 
temperature  corresponding  to  the  point,  B,  in  Fig.  48  is  called  the 
critical  solution  temperature. 

The  mutual  solubilities  of  some  pairs  of  partially  miscible 
liquids  were  found  by  Alexieeff  to  diminish  with  increasing 
temperature.  Thus,  a  mixture  of  ether  and  water,  which  is  per- 
fectly homogeneous  at  ordinary  temperatures,  becomes  turbid 
on  warming.  A  specially  interesting  pair  of  liquids  is  nicotine 
and  water.  At  ordinary  temperatures  these  liquids  are  misci- 
ble in  all  proportions.  If  the  temperature  is  raised  above  60° 
C.,  the  solution  becomes  turbid  owing  to  incomplete  miscibility. 


158 


THEORETICAL  CHEMISTRY 


100* 


Percentage  Water  in  Nicotine 


On  continuing  to  heat  the  mixture  the  mutual  solubility  of  the 
liquids  begins  to  increase,  until  at  210°  C.  they  become  com- 
pletely soluble  again.  The  solubility  relations  of  this  binary 
mixture  are  shown  in  Fig.  49.  The  closed  solubility  curve  de- 
fines the  limits  of  the  coexistence  of  two  layers,  all  points  out- 
side of  the  curve  rep- 
resenting homogeneous 
solutions. 

Vapor  Pressure  of 
Binary  Mixtures.  The 
study  of  the  vapor 
pressures  of  binary 
mixtures  of  completely 
miscible  liquids  is  of 
great  importance  in 
connection  with  the 
possibility  of  separating 
Jvthem  by  the  process  of 
distillation.  According 
to  Henry's  law,  the 
partial  pressure  of  a 
volatile  solute  in  a 
solution  is  proportional 
100*  to  its  concentration. 
Hence,  it  might  be 
expected  that  the  part- 
ial pressures  of  the  components  of  a  binary  liquid  mixture  would 
be  directly  proportional  to  their  mol  fractions.  It  has  been  shown 
by  experiment,  however,  that  only  in  the  case  of  mixtures  whose 
components  are  closely  related  chemically,  does  this  law  obtain. 
Benzene  and  toluene  afford  an  excellent  example  of  a  pair  of 
liquids  forming  binary  mixtures  in  which  the  partial  pressures 
of  the  components  conform  very  closely  to  the  law  governing  ideal 
solutions. 

The  accompanying  table,  due  to  Schmidt,*  gives  the  vapor 
pressures  of  mixtures  of  benzene  and  toluene  at  20°,  together  wfth 
the  partial  pressures,  calculated  on  the  assumption  that  these 
pressures  are  proportional  to  the  mol  fractions  of  the  two  com- 
ponents. It  will  be  seen  on  comparing  the  observed  and  calcu- 
*  Zeit.  phys.  Chem.  99,  80  (1921). 


60C 


Percentage  Nicotine  in  Water 
Fig.  49 


SOLUTIONS 


159 


lated  values  of  the  total  pressure,  that  we  are  justified  in  regard- 
ing mixtures  of  benzene  and  toluene  as  very  close  approxima- 
tions to  ideal  solutions. 


VAPOR  PRESSURES  OF  MIXTURES  OF  BENZENE 
AND  TOLUENE  AT  20° 


Mol  Fraction 
Benzene 

Total  Press. 
Obs. 

Partial  Press. 
Toluene 

Partial  Press. 
Benzene 

Total  Press. 
Calc. 

!    0.0. 

24.5 

24.5 

0.1 

30.0 

22.1 

7.7 

29.8   ' 

0.2 

36.0 

19.6 

15.3 

35.9 

0.3 

41.5 

17.2 

23.0 

40.2 

0.4 

47.0 

14.7 

30.6 

45.3 

0.5 

51.0. 

12.2 

38.2 

50.4 

0.6 

56.  Q 

9.8 

45.9 

55.7 

0.7 

62.  Q 

7.3 

53.5 

60.8 

0.8 

67.4 

5.1 

61.2 

66.3 

0.9 

74.0 

2.4 

68.8 

71.2 

1  0 

76.5 

76  5 

76  5 

In  general,  the  lack  of  agreement  between  the  observed  and 
calculated  values  of  the  vapor  pressures  of  binary  liquid  mix- 
tures is  much  greater  than  in  the  case  of  mixtures  of  benzene 
and  toluene.  The  following  table,  also  due  to  Schmidt,  gives 
the  vapor  pressures  of  mixtures  of  benzene  and 'methyl  alcohol, 
two  liquids  of  widely  divergent  character. 


VAPOR  PRESSURES  OF  MIXTURES  OF  BENZENE 
AND  METHYL  ALCOHOL  AT  20° 


Mol  Fraction 
Methyl 
Alcohol 

Total  Press. 
Obs. 

Partial  Press. 
Benzene 

Partial»Press. 
Methyl  Alcohol 
Calc. 

Total  Press. 
Calc. 

0  0 

76  5 

76.5 

76.5 

0.1 

106.5 

68.9 

9.4 

78.2 

0.2 

128.0 

61.2 

18.8 

80.0 

0.3 

139.5 

53.6 

28.2 

81.8 

0.4 

144.0 

45.9 

37.6 

83.5 

0.5 

145.5 

38.3 

47.0 

85.3 

0.6 

145.0 

30.6 

56.4 

87.0 

.7 

145.0 

23.0 

65.8 

88.8 

0.8 

141.5 

15.3 

75.8 

91.1 

0.9 

123.0 

7.7 

84.6 

92.3 

1.0      , 

94  0 

94.0 

94.0 

160 


THEORETICAL  CHEMISTRY 


The  preceding  table  serves  to  emphasize  the  fact,  that  the 
difference  between  actual  solutions  and  ideal  solutions  becomes 
more  apparent  as  the  difference  in  character  of  the  components 
becomes  more  pronounced.  While  the  cause  of  these  deviations 
is  not  known  with  certainty,  it  is  believed  that  they  may  be  due, 
either  wholly  or  in  part,  to  the  formation  of  molecular  complexes 
in  the  solution. 

The  experimental  investigations  of  Konowalow*  on  homoge- 
neous binary  mixtures  of  liquids  have  shown,  that  they  may  be 
divided  into  three  classes  as  follows: 

(1)  Mixtures  having  vapor  pressures  intermediate  between 
the  vapor  pressures  of  the  pure  components,  e.g.,  methyl  alcohol 
and  water;  (2)  mixtures  having  a  minimum  vapor  pressure  cor- 
responding to  a  definite  composition,  e.g.,  formic  acid  and  water; 

and  (3)  mixtures  having  a  max- 
imum vapor  pressure  correspond- 
ing to  a  definite  composition,  e.g., 
propyl  alcohol  and  water. 

In  considering  the  possibility  of 
separating  binary  liquid  mixtures 
belonging  to  anyone  of  these  three 
classes,  it  is  essential  to  determine 
the  composition  of  both  liquid 
and  vapor  phases.  When  a  pure 
liquid  is  boiled  the  composition  of 
the  escaping  vapor  is  the  same  as 
that  of  the  liquid  itself,  but  this  is 
only  true  under  special  conditions  when  a  binary  liquid  mixture  is 
distilled. 

A  diagrammatic  representation  of  the  vapor  pressures,  at  a 
single  temperature,  of  all  possible  mixtures  of  two  liquids  A  and 
B  belonging  to  Class  (1)  are  given  in  Fig.  50.  Since  the  vapor 
pressure  can  be  expressed  in  terms  of  the  composition  of  either 
the  liquid  or  the  vapor  phase,  it  follows  that  there  should  always 
be  two  curves  in  such  a  diagram,  one  referring  to  the  liquid  and 
the  other  to  the  vapor  phase.  In  Fig.  50  the  curves  Al'lB  and 
Avv'B,  -represent  the  variation  in  the  vapor_rjregsure  of  the  liquid 
and  vapor  phases  respectively  with  change  in  composition.  The 
vapor  phase  being  relatively  richer  in  the  component  having  the 
*  Wied.  Ann.,  14,  34  (1881). 


x' 


mol  fraction  of  B 

Fig.  50 


SOLUTIONS 


161 


higher  vapor  pressure,  it  follows  that  the  curve  Avv'B  must  lie 
nearer  to  the  ordinate  corresponding  to  pure  A  than  the  curve 
Bll'A.  By  determining  where  a  horizontal  line  cuts  the  two 
curves,  we  can  ascertain  the  equilibrium  concentrations  of  the 
two  phases.  "Thus^^jf  x  denotes  the  mol  fraction  of  B  in  the 
liquid  phase,  x'  will  bejbhe  equilibrium  concentration  of  B  in  the 
vapor  phase :  or,  if  x  aenotes  the  concentration  of  B  in  the  vapor 
phase,  x"  will  be  the  equilibrium  concentration  in  the  liquid  phase. 

In  other  words,  if  a  binary  liquid  mixture  whose  composition  is 
x  be  subjected  to  a  process  of  isothermal  distillation,  we  shall  at 
first  obtain  a  vapor  whose  composition  will  be  xr.  As  the  dis- 
tillation proceeds,  however,  the  composition  of  the  liquid  phase 
will  become  less  rich  in  the  component  B,  and  assuming  that  none 
of  the  vapor  is  allowed  to  escape,  the  last  drop  of  liquid  to  evap- 
orate will  have  the  composition  represented  by  x77"  while  the 
corresponding  composition  of  the~~vapor  phase  will  be  repre- 
sented by  x.  Mixtures  of  this  class  can  be  separated  into  their 
components  by  fractional  distillation. 

Similar  vapor  pressure  diagrams  corresponding  to  the  other 
two  classes  of  binary  liquid  mixtures  are  given  in  Figs.  51  and  52. 


mol  fraction  of  B 
Fig.    51 


mol  fraction  of  B 
Fig.    52 


Distillation  of  Binary  Mixtures.  In  the  preceding  paragraphs 
we  have  seen  that  the  varjo^_jDressure_^uxy^es  of  binary  liquid 
mixtures  at  constant  temperature"  may  be  divided  into  three 
classes.  Similarly,  when  such  mixtures  are  distilled  under  con- 
stant pressure,  we  also  distinguish  three  corresponding  classes  of 
boiling-point  curves. 

In  Class  (1)  the  boiling-points  of  all  possible  mixtures  are  inter- 


162 


THEORETICAL  CHEMISTRY 


x<"\ 


mediate  between  those  of  the  two  components.  All  mixtures 
belonging  to  Class  (2)  exhibit  a  maximum  in  their  boiling-point 
curves,  while  the  mixtures  of  Class  (3)  exhibit  a  minimum.  Each 
of  these  three  classes  of  curves  is  illustrated  diagrammatically 
in  Figs.  53  to  55.  Since,  in  binary  liquid  mixtures,  the  lower 
boiling  component  is  present  in  relatively  greater  amount  in  the 

vapor  phase,  it  follows  that  in 
each  of  the  three  diagrams,  the 
upper  curve  corresponds  to  the 
vapor  phase. 

Let  x,  Fig.  53,  represent  the 
composition  of  a  liquid  mixture 
belonging  to  Class  (1).  When 
this  mixture  is  distilled,  the  first 
portion  of  vapor  to  condense  will 
have  the  composition  x'  and,  as 
distillation  proceeds,  ^the  liquid 
phase  will  become  gradually 
poorer  in  the  component  B.  The 
vapor  phase  being  removed  by 
condensation,  the  boiling-point 
will  continue  to  rise  until  the  liquid  remaining  in  the  distilling 
flask  is  pure  A. 

Any  binary  mixture  of  this  class  can  be  more  or  less  completely 
separated  into  its  components  by  the  process  of  fractional  distilla- 
tion. For  example,  if  the  portion  of  thg-distillate  obtained  while  the 
composition  of  the  original  mixture  changes  f rom  I  to  I'  be  subjected 
to  redistillation,  the  average  composition  of  the  liquid  in  the  dis- 
tilling flask  will  be  represented  by  x" ,  and  the  initial  composition 
of  the  condensing  vapor  will  correspond  to  x"'.  In  other  words, 
in  the  process  of  fractional  distillation,  the  concentration  of  the 
component  B  in  the  distillate  has  been  increased. 

Binary  mixtures  of  Class  (2)  exhibit  a  maximum  in  their  boil- 
ing-point curves,  as  shown  at  M  in  Fig.  54.  On  distilling  a  mix- 
ture whose  initial  composition  corresponds  to  I,  the  liquid  phase 
will  become  relatively  richer  in  the  component  B,  and  the  boiling- 
point  will  continue  to  rise  until,  ultimately,  if  the  vapor  be  al- 
lowed to  condense,  the  liquid  ph^se  will  acquire  the  composition 
corresponding  to  M.  This  being  a  constant  boiling  mixture,  it 
will  behave  in  many  respects  like  a  pure  compound,  and  will 


X  X" X' 

mol  fraction  of  B 
Fig.   53 


SOLUTIONS 


163 


distil  without  change  in  composition,  provided  the  pressure  re- 
mains constant.  Similarly,  if  a  mixture  whose  initial  com- 
position corresponds  to  V  be  distilled,  the  liquid  phase  will  become 
relatively  richer  in  the  component  A  and,  finally,  will  acquire  the 
composition  of  the  mixture  having  the  maximum  boiling-point. 
It  was  thought  for  a  long  time  that  such  constant  boiling  mixtures 
were  definite  chemical  compounds  of  the  two  liquids.  Thus,  a 
mixture  of  hydrochloric  acid  and 
water  containing  20.2  per  cent 
of  acid  boils  at  110°  C.  under 
atmospheric  pressure,  The  com- 
position of  such  a  mixture  cor-  | 
responds  very  nearly  to  the  a 
formula,  HC1.8  H2O.  Roscoe*  J 
showed  that  these  mixtures  are 
not  definite  chemical  compounds 
since  the  composition  of  the  dis- 
tillate changes  when  the  distil- 
lation is  carried  out  under  dif- 
ferent pressures.  It  is  worthy 
of  emphasis  that  any  solution  of  hydrochloric  acid,  whatever 
may  be  its  concentration,  if  boiled  under  a  pressure  of  760  mm. 
for  a  sufficient  time,  will  ultimately  boil  at  110°  and  will 
contain  20.2  per  cent  HC1  by  weight.  A  partial  list  of  binary 
liquid  mixtures  having  constant  maximum  boiling-points  under 
atmospheric  pressure  is  given  in  the  following  table. 

BINARY  MIXTURES  HAVING  MAXIMUM  BOILING-POINTS 


mol  fraction  of  B 
Fig.  54 


Components 

(Pressure  =  760  mm.) 
Boiling-Point 

Composition  of  Mix- 
ture at  B.-P. 

Water  +  HNO3.. 
Water  +  HC1 

120.5° 
110 

68%  HN03 
20  2%  HC1 

Water  +  HBr.  .    . 

126 

47.5%  HBr 

Water  +  HI  

127 

57.0%  HI 

Water  +  HF  .    . 

120 

37.0%  HF 

Water  +  H.COOH       .  .  . 
CHC13  +  CHaCOCHs  .  .  . 

107.1 
64.7 

77.  9%  H.COOH 
80.0%CHC13 

A  large  number  of  pair|g|rikiuids  are  also  known  whose  boiling- 
point  curves  pass  throu^WPbiinimum,  as  shown  in  Fig.  55. 
*  Lieb.  Ann.,  116,  203  (1860). 


164 


THEORETICAL  CHEMISTRY 


mol  fraction  of  B 

Fig.  55 


These  mixtures  obviously  belong  to  Class  (3) .  Any  mixture  whose 
composition  lies  between  that  of  A  and  M,  Fig.  55,  may  be  sep- 
arated more  or  less  completely  into  pure  A  and  the  mixture  M 
by  fractional  distillation.  In  like  manner,  mixtures  whose  com- 
position lies  between  that  of  B  and  M  can  also  be.  separated  by 
fractional  distillation.  Pure  B,  however,  cannot  be  obtained  on 
distilling  a  mixture  whose  composition  lies  between  that  of  A 

and  M.  If  the  initial  composi- 
tion of  a  mixture  belonging  to 
this  class  differs  but  slightly 
from  that  corresponding  to  M, 
it  is  apparent  from  the  dia- 
gram that  the  composition  of 
the  liquid  phase  will  gradually 
change  as  the  process  of  distil- 
lation continues.  It  follows, 
therefore,  that  unless  great  care 
is  taken  to  prepare  a  mix- 
ture whose  composition  shall 
be  exactly  the  same  as  that 
of  M,  the  boiling-point  will  slowly  rise  and  the  composition 
of  the  mixture  will  undergo  a  gradual  change.  Ethyl  alcohol, 
(b.-p.  78.3°)  and  water,  (b.-p.  100°)  furnish  an  interesting 
example  of  a  pair  of  liquids  belonging  to  this  class  of  mixtures. 
The  mixture  containing  95.57%  of  alcohol  by  weight  has  a  mini- 
mum boiling-point  of  78.13°.  Any  mixture  containing  less  than 
95.57%  of  alcohol  behaves  similarly  to  mixtures  of  Class  (1),  and 
can  be  completely  separated  by  fractional  distillation  into  pure 
water  and  95.57%  ethyl  alcohol.  Theoretically,  it  should  be 
possible  to  fractionate  mixtures  having  compositions  lying  between 
95.57%  alcohol  and  pure  alcohol;  practically,  however,  the 
difference  between  the  boiling-points  of  the  minimum  boiling 
mixture  and  that  of  pure  alcohol  is  so  small,  that  the  composition 
of  all  mixtures  within  this  range  changes  with  extreme  slowness. 
We  now  turn  to  the  consideration  of  the  distillation  of  binary 
mixtures  composed  of  partially  miscible  and  immiscible  liquids. 
In  general  when  two  liquids  are  mixed,  each  lowers  the  vapor 
pressure  of  the  other,  so  that  the  vapor  pressure  of  the  mixture  is 
less  than  the  sum  of  the  vapor  pressures  of  the  components.  As 
has  already  been  pointed  out;  the  composition  of  the  two  layers 


SOLUTIONS 


165 


in  a  binary  mixture  of  partially  miscible  liquids  is  independent  of 
the  relative  amounts  of  the  components  present;  hence  the  vapor 
pressure  remains  constant  so  long  as  the  solution  remains  hetero- 
geneous. 

In  general,  the  boiling-point  curves  for  mixtures  of  two  partially 
miscible  liquids  resemble  those  shown  in  Fig.  56.  Mixtures  in 
which  the  mol  fraction  of  B  lies  between  0  and  x  and  also  between 
x'  and  1,  consist  of  a  single 
homogeneous  liquid  phase. 
On  the  other  hand,  all  mix- 
tures whose  composition  is 
included  between  x  and  x'  g 
consist  of  two  homogeneous  I 
liquid  phases.  If  a  mixture  | 
whose  composition  corre-  H 
spends  to  I  be  distilled,  the 
initial  portions  of  vapor  will 
have  the  composition  corre- 
sponding to  o.  As  distilla- 
tion proceeds  the  temperature 
will  rise  and  the  composition 
of  the  liquid  phase  will  alter,  until  eventually  nothing  but  pure  A 
will  remain  in  the  distilling  flask.  In  like  manner,  if  a  homogene- 
ous mixture  whose  composition  corresponds  to  I'  be  distilled,  the 
boiling-point  will  gradually  rise,  until  that  of  the  pure  component 
B  is  reached.  On  the  other  hand,  when  any  mixture  whose 
composition  lies  between  x  and  x'  is  dist^^f  the  boiling-point 
remains  constant  so  long  as  the  two  li)a|pWphases  are  present. 
The  composition  of  the  two  phases  is  represented  by  x  and  x' 
while  the  composition  of  the  vapor  from  each  phase  is  represented 
by  x",  this  being  the  composition  corresponding  to  the  intersection 
of  the  vapor  curves,  AC  and  BC.  If  the  process  of  distillation  is 
continued  for  a  sufficient  time,  one  of  the  two  liquid  phases  will 
disappear.  If  the  original  two-phase  liquid  mixture  was  richer 
in  component  A  than  x",  then  the  phase  represented  by  x'  will 
disappear,  and  the  boiling-point  will  increase  along  DA.  Simi- 
larly, if  the  original  mixture  was  richer  in  component  B  than 
x",  the  phase  represented  by  x  will  ultimately  disappear,  and  the 
boiling-point  will  increase  along  EB.  Numerous  examples  of 
mixtures  of  this  kind  are  known;  of  these  we  may  mention  water, 


mol  fraction  of  B 

Fig.  56 


166 


THEORETICAL  CHEMISTRY 


(b.-p.  100°)  and  isobutyl  alcohol,  (b.-p.  108.4°)  which  form  a 
two-phase  mixture  boiling  at  88.5°,  under  a  pressure  of  760  mm. 
The  relation  between  the  boiling-points  and  the  composition  of 
all  possible. mixtures  of  a  pair  of  immiscible  liquids  is  shown  dia- 
grammatically  in  Fig.  57.  The  points  D  and  E  may  be  regarded 
as  representing  the  limits  toward  which  the  points  D  and  E  in 
Fig.  56  tend,  as  the  miscibility  of  the  partially  miscible  com- 
ponents becomes  less  and  less. 
The  composition  of  the  liquid  is 
represented  by  AD,  DEand  EB, 
while  that  of  the  vapor  is  repre- 
sented by  AC  and  CB.  The 
boiling-point  of  the  two-phase 
liquid  mixture  will  remain  con- 
stant for  a  given  pressure,  and 
of  course,  will  be  less  than  that 
of  either  component,  A  or  B. 
The  composition  of  the  vapor 
phase  corresponding  to  the  point 
C  can  be  calculated  in  the  fol- 
lowing manner :  If  the  vapor  pressures  of  the  two  components 
A  and  B,  at  the  temperature  corresponding  to  C,  be  pA  and  pB, 
and  if  nA  and  nB  be  their  respective  mol  fractions,  we  may  express 
the  relation  between  vapor  pressure  and  composition  by  the 
equation, 

2^  =  5*.  (5) 

PB      nB 

If  WA  and  WB  denote  the  weights  in  grams  of  A  and  B  in  a 
known  weight  of  vapor  (distillate),  and  if  MA  and  MB  are  their 
respective  molecular  weights,  we  may  write 

nA  =  wA/MA  and  nB  =  wB/MB. 


mol  fraction  of  B 

Fig.  57 


On  substituting  these  values  in  equation  (5),  we  have 

WA_  _  pA    MA 
WB       ps    MB 


(6) 


Nitrobenzene  and  water  may  be  chosen  as  an  example  of  a  pair 
of  liquids  which  are  practically  immiscible.  Under  a  pressure 
of  760  mm.  the  mixture  boils  at  99°  C.  The  vapor  pressure  of 
water  at  this  temperature  is  733  mm.",  therefore  the  vapor  pres- 


SOLUTIONS  167 

sure  of  nitrobenzene  must  be,  760  —  733  =  27  mm.  Notwith- 
standing the  relatively  small  vapor  pressure  of  nitrobenzene  in 
the  mixture,  considerable  quantities  of  it  distil  over  with  the 
water.  It  is  this  fact  that  makes  it  possible  to  separate  liquids 
by  the  process  of  steam  distillation,  so  frequently  employed  by  the 
organic  chemist.  The  relative  weights  of  water  and  nitrobenzene 
passing  over  in  a  steam  distillation  may  be  calculated  as  fol- 
lows :  —  The  relative  volumes  of  steam  and  vapor  of  nitrobenzene 
which  distil  over,  will  be  in  the  ratio  of  their  respective  vapor 
presssures  at  the  temperature  of  the  experiment,  and  consequently 
the  relative  weights  of  the  two  liquids  which  pass  over,  will  be 
in  the  ratio,  pAdA  '  psdB)  where  pA  and  pB  denote  the  respective 
vapor  pressures  of  water  and  nitrobenzene,  and  dA  and  dB  the 
corresponding  vapor  densities.  If  WA  and  WB  denote  the  weights 
of  the  two  liquids  in  the  state  of  vapor,  then 

WA  :  WB  : :  pAdA  :  psdB) 

or,  since  vapor  density  is  proportional  to  molecular  weight,  we 
may  write 

WA  :  WB  ::  pAMA  :  pBMB. 

Substituting  in  this  proportion  the  values  given  above  for  the 
vapor  pressures  of  steam  and  nitrobenzene,  we  have 

WA  :  WB  ::  733  X  18  :  27  X  1^3 

or, 

WA  :wB  ::  13,194  :  3321. 

Thus,  the  weights  of  water  and  nitrobenzene  in  the  distillate  are 
approximately  in  the  ratio  of  4  to  1,  notwithstanding  the  fact  that 
the  ratio  of  their  vapor  pressures  at  the  boiling-point  of  the  mix- 
ture is  2'7  to  1.  If  an  organic  substance  is  not  decomposed  by 
steam,  it  is  possible  to  effect  an  appreciable  purification  by  steam 
distillation,  even  though  its  vapor  pressure  be  relatively  small. 
As  will  be  seen  from  the  above  example,  it  is  the  high  molecular 
weight  of  the  nitrobenzene  which  compensates  for  its  low  vapor 
pressure.  It  is  the  small  molecular  weight  of  water  which  renders 
it  so  suitable  for  steam  distillation. 

Solutions  of  Solids  in  Liquids.     The  solubility  of  a  solid  in  a 
liquid  is  limited,  and  is  dependent  upon  the  temperature,  the 


168  THEORETICAL  CHEMISTRY 

nature  of  the  solute  and  the  nature  of  the  solvent.  When  a 
solvent  has  taken  up  as  much  of  a  solute  as  it  is  capable  of  dis- 
solving at  a  definite  temperature,  the  solution  is  said  to  be  satu- 
rated. There  are  two  general  methods  for  the  preparation  of 
saturated  solutions:  —  (1)  An  excess  of  the  finely-divided  solute 
is  agitated  with  a  known  amount  of  the  solvent,  at  a  definite 
temperature,  until  equilibrium  is  attained;  or  (2)  the  solvent  is 
heated  with  an  excess  of  the  solute  to  a  temperature  higher  than 
that  at  which  saturation  is  required,  and  then  cooled  in  contact 
with  the  solid  solute  to  the  desired  temperature.  Both  of  these 
methods  give  equally  satisfactory  results,  provided  sufficient 
time  is  allowed  for  the  establishment  of  equilibrium,  and  provided 
the  solid  solute  is  always  present  in  excess.  The  solubility 
of  solids  has  recently  been  shown  to  be  somewhat  dependent 
upon  their  state  of  division.  Thus,  Hulett  *  has  found  that  a 
saturated  solution  of  gypsum  at  25°  C.  contains  2.080  grams  of 
CaSO4  per  liter,  whereas  when  very  finely  divided  gypsum  is 
shaken  with  this  solution,  it  is  possible  to  increase  the  content 
of  dissolved  CaSO4  to  2.542  grams  per  liter.  When  a  saturated 
solution  is  cooled,  every  trace  of  solid  solute  being  excluded,  the 
excess  of  dissolved  solid  may  not  separate.  Such  a  solution  is 
said  to  be  supersaturated. 

As  a  general  rule  the  solubility  of  solids  in  liquids  increases 
with  the  temperature,  as  shown  in  Fig.  58.  Several  exceptions 
to  this  rule  are  known,  among  which  may  be  mentioned  calcium 
hydroxide,  calcium  sulphate  above  40°  C.,  and  sodium  sulphate 
between  the  temperatures  of  33°  C.  and  100°  C. 

Solubility  curves  are  usually  continuous,  but  exceptions  to  this 
rule  are  common:  the  solubility  curve  of  sodium  sulphate  fur- 
nishes an  illustration.  The  discontinuity  in  the  solubility  curve 
of  sodium  sulphate  is  due  to  the  fact  that  we  are  not  dealing  with 
one  solubility  curve,  but  with  two  solubility  curves.  At  tempera- 
tures below  33°  C.,  the  dissolved  salt  is  in  equilibrium  with  the 
decahydrate,  Na2SO4.10  H2O,  whereas  at  temperatures  above 
33°  C.  the  dissolved  salt  is  in  equilibrium  with  the  anhydrous 
salt,  Na2SO4.  The  solubility  of  Na2SO4.10H2O  increases  with 
the  temperature,  while  the  solubility  of  Na2SO4  diminishes.  That 
we  are  actually  dealing  with  two  solubility  curves,  is  proved  by 
the  fact  that  the  solubility  curves  of  the  hydrated  and  anhydrous 
*  Jour.  Am.  Chem.  Soc.,  27,  49  (1905). 


SOLUTIONS 


169 


salts  in  supersaturated  solutions  are  continuations  of  the  corre- 
sponding curves  for  saturated  solutions,  as  shown  by  the  dotted 
curves  in  Fig.  58.  If  we  select  any  point,  such  as  p,  lying  between 
a  dotted  curve  and  a  full  curve,  it  is  apparent  that  it  represents  a 
solution  supersaturated  with  respect  to  Na^SO^lO  H2O,  but  un- 
saturated  with  respect  to  Na2SO4.  If  pure  anhydrous  sodium 
sulphate  be  shaken  with  this  solution  it  will  slowly  dissolve,  where- 
as if  a  trace  of  the  hydrated  salt  be  added,  the  solution  will  deposit 
Na2SO4.10H2O,  until  the  amount  remaining  in  solution  corre- 
sponds to  the  solubility  of  the  hydrate  at  that  temperature. 


20  40  60  80 


100 


Fig.  58 


Supersaturated  solutions  of.  some  substances  can  be  preserved 
indefinitely,  provided  all  traces  of  the  solid  phase  are  excluded. 
Such  solutions  are  called  metastable.  On  the  other  hand  there 
are  some  supersaturated  solutions  which  deposit  the  excess  of 
solid  solute  even  when  all  traces  of  it  are  excluded.  These  solu- 
tions are  termed  labile.  The  distinction  between  metastable  and 
labile  solutions  is  not  sharp.  If  a  metastable  solution  is  suffi- 
ciently cooled,  or  if  its  concentration  is  sufficiently  increased,  it 
may  be  made  to  pass  over  into  the  labile  condition.  The  eoncen- 


170  THEORETICAL  CHEMISTRY 

tration  at  which  this  transition  occurs  is  termed  the  metastable 
limit.  The  stability  of  supersaturated  solutions  has  been  shown 
by  Young  *  to  be  greatly  influenced  by  vibrations  or  sudden 
shocks  within  the  solution.  He  has  been  able  to  control  the 
amount  of  overcooling  in  a  supersaturated  solution,  by  alter- 
ing the  intensity  of  the  vibrations  due  to  the  friction  between 
glass  or  metal  surfaces  within  the  solution. 

Very  little  is  known  concerning  the  relation  between  solubility 
and  the  specific  properties  of  solute  and  solvent. 

Owing  to  the  fact  that  the  change  in  volume  resulting  from  the 
solution  of  a  solid  in  a  liquid  is  very  small,  the  effect  of  pressure 
on  the  solution  is  almost  negligible.  The  chief  factors  condition- 
ing the  change  in  solubility  due  to  increasing  pressure,  are  the 
heat  of  solution  of  the  solute  in  the  nearly  saturated  solution, 
and  the  change  in  volume  on  solidification.  Very  few  experiments 
have  been  made  to  determine  the  effect  of  pressure  on  solubility. 
It  is  stated  by  van't  Hoff  that  the  solubility  of  a  solution  of 
ammonium  chloride,  a  salt  which  expands  when  dissolved,  de- 
creases by  1  per  cent  for  160  atmospheres,  while  the  solubility  of 
copper  sulphate,  a  salt  which  contracts  when  dissolved,  increases 
by  3.2  per  cent  for  60  atmospheres. 

Solid  Solutions.  In  general,  when  a  dilute  solution  is  suffi- 
ciently cooled,  the  solvent  separates  in  the  form  of  crystals  which 
are  almost  entirely  free  from  the  solute.  When,  however,  the 
temperature  of  a  solution  of  iodine  in  benzene  is  reduced  to  the 
^freezing-point,  the  crystals  which  separate  are  found  to  contain 
iodine.  Furthermore,  the  depression  of  the  freezing-point  of  the 
solvent  is  found  to  be  less  than  that  calculated  on  the  assumption 
that  the  solvent  crystallizes  uncontaminated  with  the  solute. 
Such  solutions  were  first  studied  by  van't  Hoff.f  He  found  that 
when  the  concentration  of  such  abnormal  solutions  is  varied,  the 
ratio  of  the  amount  of  solute  in  the  liquid  solvent  to  the  amount 
of  solute  in  the  solidified  solvent  remains  constant.  Thus,  in  solu- 
tions of  iodine  in  benzene,  the  ratio  of  the  concentration  of  iodine 
in  the  liquid  phase  to  its  concentration  in  the  crystallized  ben- 
zene is  constant.  In  the  following  table,  Ci  is  the  concentration  of 
iodine  in  the  liquid  benzene,  and  c%  is  the  concentration  of  iodine 
in  the  solid  benzene. 

*  Jour.  Am.  Chem.  Soc.,  33,  148  (1911). 
t  Zeit.  phys.  Chem.,  5,  322  (1890). 


SOLUTIONS 


171 


DISTRIBUTION  OF  IODINE  BETWEEN  LIQUID  AND  SOLID 

BENZENE 


Cl 

C2                . 

C2/C1 

3.39 

2.587 
0.945 

1.279 
0.925 
0.317 

0.377 
0.358 
0.336 

It  was  van't  Hoff  who  pointed  out  the  analogy  between  the 
distribution  of  the  solute  between  the  solid  and  liquid  solvent  on 
the  one  hand,  and  the  distribution  of  a  gas  between  a  liquid  and 
the  free  space  above  it  on  the  other.  In  other  words,  the  dis- 
tribution follows  Henry's  law  for  the  solution  of  a  gas  in  a  liquid. 
Since  the  crystals  containing  both  solute  and  solvent  are  per- 
fectly homogeneous,  van't  Hoff  suggested  that  they  be  regarded 
as  solid  solutions. 

The  mixed  crystals  which  separate  from  solutions  of  isomor- 
phous  substances,  being  chemically  and  physically  homogeneous, 
may  also  be  considered  as  solid  solutions.  Many  alloys  possess 
the  properties  characteristic  of  solid  solutions;  hardened  steel, 
for  example,  being  regarded  as  a  homogeneous  solid  solution  of 
carbon  in  iron. 

One  of  the  characteristic  properties  of  a  dissolved  substance  is 
its  tendency  to  diffuse  into  the  pure  solvent.  Interesting  experi- 
ments performed  by  Roberts- Austen*  have  shown  that  even  solids 
have  the  property  of  mixing  by  diffusion.  Thus,  by  keeping  gold 
and  lead  in  contact  at  constant  temperature  for  four  years,  he 
was  able  to  detect  the  presence  of  gold  in  the  layer  of  lead  at  a 
distance  of  7  mm.  from  the  surface  of,  separation.  Many  other 
instances  of  diffusion. in  solids  have  been  observed. f 

Instances  of  gases  and  liquids  dissolving  in  solids  are  also 
known.  Thus  platinum,  palladium,  charcoal  and  other  sub- 
stances have  the  property  of  taking  up  large  volumes  of  hydrogen. 
This  phenomenon,  known  as  occlusion,  is  but  little  understood. 
It  was  suggested  by  van't  Hoff  that  when  hydrogen  dissolves  in 
palladium,  we  are  really  dealing  with  two  solid  solutions,  one  a 

*  Proc.  Roy.  Soc.,  67,  101  (1900). 

f  See  Report  on  Diffusion  in  Solids,  by  C.  H.  Desch,  Chem.  News,  106, 
153  (1912). 


172  THEORETICAL  CHEMISTRY 

solution  of  hydrogen  in  palladium,  and  the  other  a  solution  of 
palladium  in  solid  hydrogen,  the  system  being  analogous  to  that 
of  two  partially  miscible  liquids. 

Certain  natural  silicates,  the  so-called  zeolites,  are  transparent 
and  homogeneous.  Since  they  contain  varying  quantities  of 
water  they  may  be  regarded  as  examples  of  solutions  of  liquids 
in  solids.  This  classification  is  further  justified  by  the  fact,  that 
portions  of  the  water  may  be  removed  and  replaced  by  other 
substances,  such  as  alcohol,  with  apparently  no  change  in  the 
transparency  or  homogeneity  of  the  mineral. 

REFERENCE 

Stoichiometry.     Young.     Chapters  XII  to  XIV  inclusive. 

/  PROBLEMS 

*\J/  A  solution  of  potassium  chloride  contains  245.7  grams  of  salt  in 
1000  grams  of  water,  and. the  density  of  the  solution  at  21°  is  1.1310. 
Calculate  the  weight-  and  volume-formal  concentration  of  the  solution. 
Also  calculate  the  mol  fractions  of  the  two  components  of  the  solution. 
s  2.  A  binary  mixture  of  chloroform  and  acetone  contains  30%  of  the 
former  to  70%  of  the  latter.  Calculate  the  mol  fractions  of  the  com- 
ponents. 

3.  A  10%  aqueous  solution  of  sulphuric  acid  has  a  density  of  1.0661 
at  20°.     Calculate  its  weight-normal  and  volume-molal  concentrations. 

4.  The  density  of  a  solution  of  barium  chloride  containing  1  mol  of 
salt  to  100  mols  of  water  is  1.0976  at  15°.     What  per  cent  of  barium  chlor- 
ide does  the  solution  contain?    What  is  its  volume  normality  with  respect 
to  barium  chloride? 

5.  The  density  of  a  20%  aqueous  solution  of  sodium  chloride  is  1.1473 
at  20°.     Calculate  its  weight-formal  concentration.     Express  the  con- 
centrations of  the  two  components  in  mol  fractions. 

6.  2.3  liters  of  hydrogen  -under  a  pressure  of  78  cm.  of  mercury,  and 
5.4  liters  of  nitrogen  at  a  pressure  of  46  cm.  were  introduced  into  a  vessel 
containing  3.8  liters  of  carbon  dioxide  under  a  pressure  of  27  cm.     What 
wasthe  pressure  of  the  mixture?  Ans.  140  cm  of  mercury. 

OQ  Air  is  composed  of  20.9  volumes  of  oxygen  and  79.1  volumes  of 
nitrogen.  At  15°  C.  water  absorbs  0.0299  volumes  of  oxygen  and  0.0148 
volumes  of  nitrogen,  the  pressure  of  each  being  that  of  the  atmosphere. 
Calculate  the  composition  of  the  mixture  of  gases  absorbed  by  the  water. 

Ans.  34.8%  by  vol.  of  0,  and  65.2%  by  vol.  of  N. 

^>8.  The  vapor  pressure  of  ethyl  ether  at  20°  is  442.36  mm.  and  that 
of  benzene  is  74.66  mm.  Calculate  the  partial  pressure  of  each  in  a  solu- 
tion in  which  the  mol  fraction  of  ether  is  0.4,  and  compute  the  total  vapor 


SOLUTIONS 

pressure.     The  observed  value  of  the  vapor  pressure  is  222.5  mm.    What 
isjjje  per  cent  error  in  the  calculated  pressure? 

At  40°  the  vapor  pressures  of  acetone  and  chloroform  are  420  mm.  * 
and  366  mm.  respectively.     Calculate  the  total  vapor  pressure ^f  a  mix- 
ture in  which  the  in^l  fraction  pj^phlornfnrm  i"i  Orfi     The  observed  value 
of  the  vapor  pressure  is  318  mm.     Also  calculate  the  per  cent  error  in  the 
calculated  value  of  the  total  pressure. 

<£lO^)The  vapor  pressures  of  benzene  and  carbon  disulphide  at  20D,  40°, 
60°  and  80°  are  as  follows : 

20°  40°  60°  80° 

C6H6        76.5mm.         185mm.        394mm.          755mm. 
CS,        297  617  1165  2030 

Calculate  the  total  vapor  pressure  at  each  of  the  above  temperatures 
for  a  mixture  containing  one  mol  of  benzene  in  one  mol  of  carbon  disul- 
phide.  The  observed  values  of  the  total  pressure  at  the  above  temper- 
atures are  respectively  as  follows:  206  mm.,  432  mm.,  838  mm.,  and  1475 
mm.  Calculate  the  per  cent  error  for  each  temperature.  Is  the  solution 
more  nearly  ideal  in  its  behavior  at' the  highest  or  lowest  temperature? 
(JjLjThe  vapor  pressure  of  the  immiscible  liquid  system,  aniline- water, 
is  760  mm.  at  98°  C.  The  vapor  pressure  of  water  at  that  temperature  is 
707  mm.  What  fraction  of  the  total  weight  of  the  distillate  is  aniline. 

Am.  0.28 

12.  The  boiling-point  of  the  immiscible  liquid  system,  naphthalene- 
water,  is  98°  C.  under  a  pressure  of  733  mm.  The  vapor  pressure  of 
water  at  98°  C.  is  707  mm.  Calculate  the  proportion  of  naphthalene  in 
the  distillate.  Ans.  0.207. 


CHAPTER  VIII 
DILUTE   SOLUTIONS   AND   OSMOTIC  PRESSURE 

Osmotic  Pressure.  In  the  preceding  chapter  reference  was 
made  to  the  fact  that  diffusion  is  a  characteristic  property  of  solu- 
tions. If  a  few  cubic  centimeters  of  a  concentrated  solution  of 
cane  sugar  are  placed  at  the  bottom  of  a  tall  cylinder,  and  water 
is  added,  care  being  taken  to  prevent  mixture,  the  sugar  immedi- 
ately begins  to  diffuse  into  the  water,  the  process  continuing  until 
the  concentration  of  sugar  is  the  same  throughout  the  liquid. 
The  sugar  molecules  move  from  a  region  of  high  concentration 
to  a  region  of  low  concentration,  the  rate  of  diffusion  being  rela- 
tively slow  owing  to  the  viscosity  of  the  mpdium.  A  similar 
process  is  encountered  in  the  study  of  gases,  but  the  rate  of  gas- 
eous diffusion  is  extremely  rapid.  In  terms  of  the  kinetic  theory, 
the  movement  of  the  molecules  of  a  gas  from  regions  of  high 
concentration  to  regions  of  low  concentration,  is  to  be  considered 
as  due  to  the  pressure  of  the  gas.  By  analogy,  we  may  regard  the 
process  of  diffusion  in  solutions  as  a  manifestation  of  a  driving 
force,  known  as  the  osmotic  pressure. 

Semi-permeable  Membranes.  The  use  of  a  semi-permeable 
membrane  for  the  measurement  of  the  partial  pressure  of  nitrogen 
in  a  mixture  of  nitrogen  and  hydrogen,  has  already  been  explained. 
A  similar  method  may  be  employed  for  the  measurement  of  the 
osmotic  pressure  of  a  solution,  provided  a  suitable  semi-permeable 
membrane  can  be  found.  Such  a  membrane  must  prevent 
the  passage  of  the  molecules  of  solute  and  must  also  be  readily 
permeable  to  the  molecules  of  solvent;  it  must  exert  a  selective 
action  on  solute  and  solvent.  If  a  solution  is  separated  from  the 
pure  solvent  by  a  semi-permeable  membrane,  diffusion  of  the  solute 
is  no  longer  possible.  Since  equilibrium  of  the  system  can  only 
be  attained  when  the  concentrations  on  both  sides  of  the  mem- 
brane are  equal,  it  follows  that  the  solvent  must  pass  through 
the  membrane  and  dilute  the  more  concentrated  solution.  A 
number  of  semi-permeable  membranes  have  been  discovered 

174 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE          175 

which  are  readily  permeable  to  water  and  nearly,  if  not  entirely, 
impermeable  to  most  solutes.  About  the  middle  of  the  eight- 
eenth century  Abbe  Nollet  discovered  that  certain  animal  mem- 
branes are  permeable  to  water  but  not  to  alcohol. 

Artificial  semi-permeable  membranes  were  first  prepared  by 
M.  Traube.*  If  a  glass  tube,  provided  with  a  rubber  tube  and 
pinch-cock,  be  partially  filled,  by  suction,  with  a  solution  of 
copper  sulphate,  and  then  immersed 
in  a  solution  of  potassium  ferrocyanide, 
a  thin  film  of  copper  ferrocyanide  will 
be  formed  at  the  junction  of  the  two 
solutions.  When  the  film  has  once 
been  formed,  further  precipitation  of 
copper  ferrocyanide  will  cease,  the 
solutions  on  either  side  of  the  film 
remaining  clear.  Traube  showed  that 
this  membrane  is  semi-permeable.  He 
also  showed  that  a  number  of  other 
gelatinous  precipitates  possess  the 
property  of  semi-permeability.  A  mem- 
brane formed  in  the  above  manner  is 
easily  ruptured  and  is  wholly  inad- 
equate for  quantitative  or  even  qual- 
itative experiments.  Pfefferf  devised 
a  method  for  strengthening  the  mem- 
brane. By  depositing  the  precipitate 
in  the  walls  of  a  porous  clay  cup, 
the  area  of  unsupported  membrane 
is  greatly  diminished  and  its  resist- 
ing power  correspondingly  increased. 
Pfeffer  directs  that  the  cup  to  be  used  for  this  purpose  must 
be  thoroughly  washed,  and  its  walls  allowed  to  become  com- 
pletely permeated  with  water.  The  cup  is  then  filled  to 
the  top  with  a  solution  of  copper  sulphate,  containing  2.5 
grams  per  liter,  and  allowed  to  stand  for  several  hours  in  a 
solution  of  potassium  ferrocyanide,  containing  2.1  grams  per 
liter.  The  two  solutions  diffuse  through  the  walls  of  the  cup, 


Fig.  59 


*  Archiv.  fur  Anat.  und  Physiol.,  p.  87  (1867). 
t  Osmotische  Untersuchungen,  Leipzig:,  1877. 


176 


THEORETICAL  CHEMISTRY 


and  on  meeting,  deposit  a  thin  membrane  of  copper  ferrocyanide. 
When  precipitation  is  complete,  the  cup  is  thoroughly  washed 
and  soaked  in  water.  The  cup  is  then  filled  to  the  top  with  a 
solution  of  cane  sugar,  and  a  rubber  stopper,  fitted  with  a  long 
glass  tube  of  narrow  bore,  is  inserted,  care  being  taken  to  exclude 
air-bubbles.  The  stopper  is  then  made  fast  with  a  suitable 

cement,  and  the  cup  completely 
immersed  in  a  beaker  of  water. 
The  completed  apparatus  is  shown 
in  Fig.  59.  If  the  formation  of  the 
membrane  has  been  successful,  the 
level  of  the  liquid  in  the  vertical 
glass  tube  will  slowly  rise  and 
eventually  will  attain  a  height  of 
several  meters.  If  the  membrane 
is  sufficiently  strong  and  no  leaks 
develop,  the  passage  of  water 
through  the  membrane  will  con- 
tinue until  the  hydrostatic  pressure 
of  the  column  of  liquid  in  the  tube 
is  great  enough  to  overcome  the 
tendency  of  the  water  to  force  its 
way  into  the  sugar  solution.  As  a 
general  rule,  the  membrane  be- 
comes ruptured  before  equilibrium 
is  attained. 

Measurement  of  Osmotic  Pres- 
sure. The  first  direct  measure- 
ments of  osmotic  pressure  were 
made  by  Pfeffer.  His  experiments 
deserve  brief  consideration,  since 
the  results  obtained  furnish  the 
basis  of  the  modern  theory  of  solu- 
tion. The  cell  used  was  similar 
to  that  described  above,  but  in- 
stead  of  employing  a  vertical  glass 
tube  as  a  manometer,  the  cup  was 
connected,  as  shown  in  Fig.  60,  with  a  closed  mercury  manom- 
eter. The  substitution  of  the  closed  for  the  open  manometer  is 
necessitated  by  the  fact,  that  with  an  open  manometer  so  much 


go 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE          177 

water  entered  the  cell  that  the  concentration  of  the  solution 
became  appreciably  diminished,  and  the  pressure  actually  meas- 
ured corresponded  to  a  solution  of  smaller  concentration  than 
that  introduced  into  the  cell.  With  the  closed  manometer, 
when  a  trace  of  water  has  entered  the  cell,  sufficient  pressure 
is  developed  to  prevent  the  further  entrance  of  more  water. 
Pfeffer  calculated  that  with  a  cell,  the  capacity  of  which  was 
16  cc.,  the  volume  of  water  entering  before  equilibrium  was 
attained,  did  not  exceed  0.14  cc.  In  his  experiments,  Pfeffer 
determined  the  density  of  the  cell  contents  before  and  after 
measurement'  of  the  osmotic  pressure,  and  thus  applied  a 
correction  for  any  change  in  concentration  which  might  have 
taken  place.  With  this  apparatus  he  made  numerous  meas- 
urements of  the  osmotic  pressures  of  different  solutions,  the 
entire  apparatus  being  immersed  in  a  constant-temperature 
bath. 

With  solutions  of  cane  sugar  he  obtained  the  results  given 
in  the  accompanying  table,  where  C  denotes  the  percentage  con- 
centration of  the  solution,  and  P  the  corresponding  osmotic  pres- 
sure, expressed  in  centimeters  of  mercury.  The  temperature 
varied  from  13.5°  C.  to  14.7°  C. 


RELATION  OF  OSMOTIC  PRESSURE  TO  CONCENTRATION 


C 

P 

P/C 

1 

53.5 

53.5 

2 

101.6 

50.8 

4 

208.2 

52.0 

6 

307.5 

51.2 

It  is  evident  from  these  results,  that  the  osmotic  pressure  is 
proportional  to  the  concentration  of  the  solution,  since  P/C  is 
approximately  constant.  The  deviations  from  constancy  in  the 
ratio  of  pressure  to  concentration  may  be  ascribed  to  experimental 
errors,  since  the  difficulties  involved  in  these  measurements  are 
very  great.  Pfeffer  also  studied  the  influence  of  temperature  on 
osmotic  pressure,  and  showed  that  as  the  temperature  is  raised 
the  pressure  increases.  The  following  table  gives  his  results  for 
a  1  per  cent  solution  of  cane  sugar. 


178 


THEORETICAL  CHEMISTRY 


RELATION  OF  OSMOTIC  PRESSURE  TO  TEMPERATURE 


Temperature. 

Osmotic  Pressure. 

6°.  8 

cm. 

50.5 

13°.  2 

52.1 

14°.  2 

53.1 

22°.  0 

54.8 

36°.  0 

56.7 

A                                         B 

3                          4 

Osmotic  Pressure  and  the  Nature  of  the  Membrane.  Pfeffer 
investigated  the  effect  of  the  nature  of  the  membrane  on  osmotic 
pressure.  In  addition  to  copper  ferrocyanide,  he  used  membranes 
of  calcium  phosphate  and  Prussian  blue.  His  results  seemed  to 
indicate  that  the  magnitude  of  the  osmotic  pressure  developed, 
was  dependent  upon  the  nature  of  the  membrane  used. 

The  variations  observed  have  since  been  shown,  however,  to 
have  been  due  to  leakage  of  the  calcium  phosphate  and  Prussian 
blue  membranes,  the  copper  ferrocyanide  membrane  being  the  only 
one  which  was  capable  of  withstanding  the  pressure.  Ostwald  * 
has  demonstrated  theoretically  that  osmotic  pressure  must  be  in- 
dependent of  the  na- 
ture of  the  membrane 
employed  in  measuring 
it.  Thus,  let  A  and  B, 
in  Fig.  61,  represent  two 
different  semi-permeable 
membranes  placed  in  a  glass  tube  of  wide  bore.  Let  us 
imagine  the  space  between  the  two  membranes  to  be  filled 
with  a  solution,  and  the  tube  immersed  in  a  vessel  of  water. 
If  the  osmotic  pressures  developed  at  A  and  B  are  pi  and  p2 
respectively,  and  p2  is  less  than  .pit  then  water  will  pass 
through  A  until  the  pressure  pi  is  reached.  Since  the  pres- 
sure at  B  only  reaches  the  value  p2,  however,  the  pressure 
Pi  can  never  be  attained,  and  a  steady  stream  of  water  from 
A  to  B,  under  the  pressure  pi  —  p2,  will  result.  This,  however, 
would  be  a  perpetual  motion,  and  since  this  is  impossible,  the 
osmotic  pressures  at  the  two  membranes  must  be  the  same. 

Theoretical  Value  of  Osmotic  Pressure.  The  physico-chem- 
ical significance  of  Pfeffer's  results  was  first  perceived  by  van't 

*  Lehrb.  d.  allg.  Chem.,  I.,  p.  662. 


Fig.  61 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE         179 


Hoff.*  In  a  remarkably  brilliant  paper,  he  pointed  out  the 
existence  of  a  striking  parallelism  between  the  properties  of  gases 
and  the  properties  of  dissolved  substances. 

We  have  already  called  attention  to  the  analogy  between  osmotic 
pressure  and  gas  pressure:  we  now  proceed  to  trace  the  connec- 
tion between  osmotic  pressure,  volume  and  temperature,  as  first 
pointed  out  by  van't  Hoff.  Pfeffer's  experiments  showed  that  at 
constant  temperature,  the  ratio,  P/C,  is  constant  for  any  one 
solute.  Since  -the  concentration  varies  inversely  as  the  volume 
in  which  a  definite  amount  of  solute  is  dissolved,  we  obtain,  by 
substituting  1/V  for  C,  the  equation,  PV  =  constant,  which  is 
plainly  the  analogue  of  the  familiar  equation  of  Boyle  for  gases. 
An  examination  of  Pfeffer's  data  for  osmotic  pressures  at  differ- 
ent temperatures,  convinced  van't  Hoff  that  the  law  of  Gay- 
Lussac  is  also  applicable  to  solutions.  In  the  following  table, 
the  osmotic  pressures  in  atmospheres  for  a  1  per  cent  solution 
of  cane  sugar  at  different  temperatures  are  recorded,  together 
with  the  pressures  calculated  on  the  assumption  that  the  osmotic 
pressure  is  directly  proportional  to  the  absolute  temperature. 

VARIATION   OF  OSMOTIC  PRESSURE  WITH  TEMPERATURE 


Temperature. 

P  (obs.). 

P  (calc.). 

6°.  8 

0.664 

0  665 

13°.  7 

0.691 

0.681 

15°.  5 

0.684 

0.686 

22°.  0 

0.721 

0.701 

32°.  0 

0.716 

0  725 

36°.  0 

0.746 

0.735 

Since  the  laws  of  Boyle  and  Gay-Lussac  are  both  applicable, 
we  may  write  an  equation  for  dilute  solutions  corresponding  to 
that  already  derived  for  gases,  or 

PV  =  R'T,  (1) 

in  which  P  is  the  osmotic  pressure  of  a  solution  containing  a  defi- 
nite weight  of  solute  in  the  volume,  7,  of  solution,  T  being  the 
absolute  temperature  of  the  solution  and  R'  a  constant  corre- 
sponding to  the  molecular  gas  constant. 

The  molecular  gas  constant  R  has  already  been  evaluated  and 

*  Zeit.  phys.  Chem.,  i,  481  (1887). 


180  THEORETICAL  CHEMISTRY 

has  been  found  to  be  equal  to  0.0821  liter-atmosphere.  Making 
use  of  Pfeffer's  data,  van't  Hoff  calculated  the  value  of  Rr  in 
the  above  equation,  in  the  following  manner:  the  osmotic  pres- 
sure of  a  1  per  cent  solution  of  cane  sugar  at  0°  C.  is  0.649 
atmosphere,  and  since  the  concentration  of  the  solution  is  1  per 
cent,  the  volume  of  solution  containing  1  mol  of  sugar,  will  be 
34,200  cc.  or  34.2  liters.  Substituting  these  values  in  the  equa- 
tion, we  have 

PV      0.649  X  34.2 
R    =  -Tfr  =  -   -7^5  --  =  0.0813  liter-atmos., 


a  value  which  is  nearly  the  same  as  that  of  the  molecular  gas 
constant,  R.  The  equality  of  R  and  Rr  leads  to  a  conclusion  of 
the  greatest  importance,  as  was  pointed  out  by  van't  Hoff,  viz., 
"  the  osmotic  pressure  exerted  by  any  substance  in  solution  is  ike  same 
as  it  would  exert  if  present  as  a  gas  in  the  same  volume  as  that  occu- 
pied by  the  solution,  provided  that  the  solution  is  so  dilute  that  the 
volume  occupied  by  the  solute  is  negligible  in  comparison  with  that 
occupied  by  the  solvent."  It  should  be  remembered  that  we  are 
not  justified  in  concluding  from  this  proposition  that  osmotic 
pressure  and  gaseous  pressure  have  a  common  origin.  While 
the  origin  of  osmotic  pressure  may  be  kinetic,  it  is  also  conceivable 
that  it  may  result  from  the  mutual  attraction  of  solvent  and 
solute,  or  that  it  may  bear  some  relation  to  the  surface  tension 
of  the  solution.  Up  to  the  present  time  no  wholly  satisfactory 
explanation  of  the  cause  of  osmotic  pressure  has  been  advanced. 

Just  as  1  mol  of  gas  at  0°  C.  and  760  mm.  pressure  occupies  a 
volume  of  22.4  liters,  so  when  1  mol  of  a  substance  is  dissolved, 
and  the  solution  diluted  to  22.4  liters  at  0°  C.,  it  will  exert  an 
osmotic  pressure  of  1  atmosphere.  In  other  words,  molar  weights, 
or  quantities  proportional  to  molar  weights,  of  different  substances, 
when  dissolved  in  equal  volumes  of  the  same  solvent  exert  the  same 
osmotic  pressure.  If  we  deal  with  n  mols  of  solute  instead  of  1 
mol  the  general  equation  becomes, 

PV  =  nRT.  (2) 

But  n  =  g/M,  where  g  is  the  number  of  grams  of  solute  per  liter, 
and  M  is  its  molecular  weight.  Substituting  in  the  preceding 
equation,  we  have 

PV  =  g/M  -  RT, 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE          181 
or, 

(3) 


Since  P,  V,  g,  R  and  T  are  all  known,  M  can  be  calculated.  The 
direct  measurement  of  the  osmotic  pressure  of  a  solution  does  not 
afford  a  practical  method  for  the  determination  of  the  molecular 
weight  of  a  dissolved  substance,  because  of  the  experimental 
difficulties  involved  and  the  time  required  for  the  establishment 
of  equilibrium.  There  are  other  and  simpler  methods  for  deter- 
mining molecular  weights  in  solution,  based  upon  certain  proper- 
ties of  solutions  which  are  proportional  to  their  respective  osmotic 
pressures. 

The  Direct  Measurement  of  Osmotic  Pressure.  It  is  only 
within  the  past  twenty  years  that  the  investigations  of  Pfeffer 
have  been  confirmed  and  extended  by  the  elaborate  and  systematic 
experiments  of  Morse  and  his  co-workers,*  on  the  direct  measure- 
ment of  osmotic  pressure.  The  most  important  of  the  improve- 
ments introduced  by  Morse  are  the  following:  —  (1)  the  improve- 
ment of  the  quality  of  the  membrane;  (2)  the  improvement  of 
the  connection  between  the  cell  and  the  manometer,  and  (3)  the 
improvement  of  the  means  of  accurately  measuring  the  pressure. 
The  membrane  of  copper  ferrocyanide  was  deposited  electrolytic- 
ally.  After  thorough  washing  and  soaking  in  water,  the  porous 
cup,  made  from  specially  prepared  clay,  was  filled  with  a  solution 
of  potassium  ferrocyanide  and  immersed  in  a  solution  of  copper 
sulphate.  An  electric  current  was  then  passed  from  a  copper 
electrode  in  the  solution  of  copper  sulphate,  to  a  platinum  electrode 
immersed  in  the  solution  of  potassium  ferrocyanide.  This  drove 
the  copper  and  ferrocyanide  ions  toward  each  other,  and  the 
membrane  of  copper  ferrocyanide  was  thus  formed  in  the  walls 
of  the  cup.  The  passage  of  the  current  was  continued  until  the 
electrical  resistance  reached  a  value  of  about  100,000  ohms. 
The  cell  was  rinsed,  and  soaked  in  water  for  several  hours, 
and  then  the  electrolytic  treatment  was  repeated  until  the  elec- 
trical resistance  attained  a  maximum  value.  A  solution  of  cane 
sugar  was  now  introduced  into  the  cell,  which,  after  connecting 
with  the  manometer,  was  immersed  in  water.  When  the  pres- 
sure had  attained  its  maximum  value,  the  apparatus  was  disman- 
tled and  the  cell,  after  thorough  washing  and  soaking  in  water, 
*  Carnegie  Inst.  Publ.  198,  (1914). 


182 


THEORETICAL  CHEMISTRY 


was  again  subjected  to  the  electrolytic  process  of  membrane  forma- 
tion. In  this  manner  the  weak  places  in  the  membrane  were  re- 
paired, and  by  continued  repetition  of  the  treatment,  a  membrane 
possessing  maximum  resistance  was  ultimately  obtained. 

The  following  table,  taken  from  the  papers  of  Morse  and  his 
students,  shows  that  when  concentrations  are  expressed  on  the 
weight-molar  basis,  there  is  direct  proportionality  between  osmotic 
pressure  and  concentration  over  a  comparatively  wide  range  of 
both  concentration  and  temperature. 

OSMOTIC  PRESSURES  OF  SUCROSE  SOLUTIONS  IN 
ATMOSPHERES 


0° 

20° 

40° 

60° 

Av. 

Cone. 

Ratio 

Press. 

Ratio 

Press. 

Ratio 

Press. 

Ratio 

Press. 

Ratio 

0.1 

2.462 

1.106 

2.590 

1.130 

2.560 

0.998 

2.717 

1.000 

1.06 

0.2 

4.723 

1.065 

5.064 

1.060 

5.163 

1.012 

5.438 

1.001 

1.03 

0.4 

9.443 

1.060 

10.137 

1.060 

10.599 

1.037 

10.866 

1.000 

1.04 

0.6 

14.381 

1.077 

15.388 

1.071 

16.146 

1.053 

16.535 

1.015 

1.05 

0.8 

19.476 

1.091 

20.905 

1.093 

21.806 

1.068 

22.327 

1.025 

1.07 

1.0 

24.826 

1.130 

26.638 

1.130 

27.701 

1.085 

28.367 

1.045 

1.10 

In  the  columns  headed  "  Ratio  ",  are  given  the  ratios  of  the  meas- 
ured values  of  osmotic  pressure  to  the  corresponding  calculated 
values  of  gas  pressure. 

The  investigations  of  Morse  and  his  co-workers  may  be  sum- 
marized thus:  —  (1)  the  law  of  Boyle  is  applicable  to  dilute  solu- 
tions, provided  the  concentration  is  referred  to  1000  grams  of 
solvent  and  not  to  1  liter  of  solution;  (2)  the  law  of  Gay-Lussac 
is  also  applicable  to  dilute  solutions,  that  is,  the  temperature 
coefficients  of  osmotic  pressure  and  gas  pressure  are  equal,  and 
(3)  the  small  departures  from  the  theoretical  values  of  the  osmotic 
pressures  may  be  traced  to  hydra tion  of  the  solute. 

Direct  measurements  of  the  osmotic  pressure  of  concentrated 
solutions  of  cane  sugar,  dextrose  and  mannite  have  also  been  made 
by  the  Earl  of  Berkeley  and  E.  G.  J.  Hartley.*  The  method 
employed  by  these  investigators  differs  from  that  of  Pfeffer 
or  Morse,  in  that  the  tendency  of  water  to  pass  through  the 
semi-permeable  membrane  is  offset  by  the  application  of  a  counter 

*  Proc.  Roy.  Soc.,  73,  436  (1904);   Trans.  Roy.  Soc.  A.,  206,  481  (1906). 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE          183 


pressure  to  the  solution.  A  membrane  of  copper  ferrocyanide 
is  deposited  electrolytically  very  near  the  outer  surface  of  a  tube 
of  porous  porcelain.  This  tube  is  placed  co-axially  within  a  large 
cylindrical  vessel  of  gun  metal,  an  absolutely  tight  joint  between 
the  two  being  secured  by  an  ingenious  system  of  dermatine  rings 
and  clamps.  The  open  ends  of  the  porcelain  tube  are  closed  by 
rubber  stoppers  fitted  with  capillary  tubes  bent  at  right  angles, 
one  of  the  latter  being  provided  with  a  glass  stop-cock.  When  a 
determination  of  osmotic  pressure  is  to  be  made,  the  apparatus  is 
placed  in  a  horizontal  position  and  water  is  introduced  into  the 
porcelain  tube,  completely  filling  it  and  the  connecting  capillary 
tubes  up  to  a  certain  level.  The  gun  metal  vessel  is  then  filled 
with  the  solution,  and  connected  with  an  auxiliary  apparatus  by 
means  of  which  a  gradually  increasing  hydrostatic  pressure  can 
be  applied.  If  no  pressure  is  applied  to  the  solution,  water  will 
pass  through  the  semi-permeable  membrane  into  the  solution,  and 
the  level  of  the  water  in  the  capillary  tubes  will  fall.  In  carrying 
out  a  measurement,  therefore,  as  soon  as  the  solution  is  introduced 
into  the  gun  metal  vessel,  hydrostatic  pressure  is  applied,  the  mag- 
nitude of  the  pressure  being  so  adjusted  as  to  counterbalance  the 
osmotic  pressure  of  the  solution.  The  level  of  the  water  in  the 
capillary  tubes  serves  to  indicate  the  relative  magnitudes  of 
the  osmotic  and  hydrostatic  pressures.  When  the  level  of  the 
water  in  the  capillary  tubes  remains  constant,  the  two  pressures 
are  in  equilibrium.  The  following  table  gives  the  values  of  the 
equilibrium  pressures  of  solutions  of  cane  sugar,  at  0°  C.  It 
must  be  remembered  that  when  the  two  pressures  are  in  equi- 
librium, there  is  always  a  pressure  of  one  atmosphere  on  the  solvent. 

OSMOTIC  PRESSURE  OF  SUCROSE  SOLUTIONS  IN 
ATMOSPHERES 

(Temperature  0°) 


Cone.  gm.  per 
Liter. 

Press.  (Obs.) 

Press.  (Calc.) 

180.1 

13.95 

13.95 

300.2 

26.77 

28.74 

420.3 

43.97 

32.55 

540.4 

67.51 

41.85 

660.5 

100.78 

51.16 

750.6 

133.74 

58.14 

184 


THEORETICAL  CHEMISTRY 


As  will  be  seen,  the  pressures  developed  in  the  more  concen- 
trated solutions  are  enormous,  and  it  is  a  surprising  fact,  that  even 
in  cases  where  the  highest  pressures  were  measured,  hardly  a 
trace  of  sugar  was  found  in  the  pure  solvent,  proving  that  over  the 
entire  range  of  pressures  the  membrane  retained  its  property  of 
semi-permeability.  The  figures  in  the  third  column  are  calculated 
on  the  assumption  that  there  is  direct  proportionality  between 
osmotic  pressure  and  concentration.  It  is  apparent  that  in  every 
case  the  observed  osmotic  pressure  is  greater  than  the  calculated. 
Even  when  the  concentrations  are  expressed  on  the  weight-molal 
basis,  as  recommended  by  Morse,  the  osmotic  pressure  increases 
more  rapidly  than  the  concentration. 


180 


100 


80 


60 


140 


100  200  300  400  600 

Grams  Cane  Stigar  per  liter  of  Solution 

Fig.  62 


700 


This  is  well  shown  in  the  accompanying  diagram,  Fig.  62,  due 
to  the  Earl  of  Berkeley.  In  this  diagram,  the  osmotic  pressures 
of  solutions  of  cane  sugar  are  plotted  against  concentrations. 
Curve  A  represents  the  actually  observed  osmotic  pressures; 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE          185 


curve  C  is  traced  on  the  assumption  that  osmotic  pressure  may  be 
calculated  from  the  equation,  PV  =  RT,  where  F  denotes  the 
volume  of  solvent  containing  1  mol  of  cane  sugar;  and  curve  B, 
a  straight  line,  is  drawn  on  the  assumption  that  osmotic  pressure 
may  be  calculated  from  the  equation,  PV  =  RT,  where  F  repre- 
sents the  volume  of  solution  containing  1  mol. 

While  the  theoretical  and  observed  values  of  the  osmotic  pres- 
sure are  approximately  equal  in  the  more  dilute  solutions,  it  is 
obvious  that  the  observed  values  of  the  osmotic  pressure  of  the 
concentrated  solutions  are  always  greater  than  the  calculated 
values,  even  when  the  calculation  is  made  on  the  assumption  that 
F,  in  the  equation  PV  =  RT,  is  the  volume  of  the  solvent.  The 
abnormally  high  osmotic  pressures  observed  by  the  Earl  of  Berk- 
eley have  been  discussed  by  Callendar  *  who  suggests  hydra- 
tion  of  the  solute  as  a  probable  cause. 
He  shows,  that  if  5  molecules  of  water 
are  assumed  to  be  associated  with  each 
molecule  of  cane  sugar  in  the  most  concen- 
trated solutions  studied  by  the  Earl  of 
Berkeley,  the  discrepancy  between  the  ob- 
served and  calculated  values  of  the  osmotic 
pressure  disappears. 

Frazer  and  Myrickf  have  recently  devised 
an  ingenious  modification  of  the  method 
originated  by  Morse.  By  means  of  this  new 
apparatus  equilibrium  ca'n  be  established 
within  a  few  hours,  and  osmotic  pressures 
of  at  least  270  atmospheres  can  be  meas- 
ured. These  features  make  it  possible  to 
determine  the  osmotic  pressures  of  concen- 
trated solutions  with  a  high  degree  of  ac- 
curacy and  also  suggest  the  possibility  of 
determining  the  osmotic  pressures  of  electro- 
lytes. In  this  new  form  of  apparatus,  a  dia- 
gram of  which  is  shown  in  Fig.  63,  the  mem- 
brane is  deposited  on,  and  in  the  pores  near,  j?ig.  53 
the  outer  surface  of  the  cell.  The  cell  is  fast- 
ened inside  of  a  hollow  bronze  cylinder  to  which  the  manometer  is 

*  Proc.  Roy.  Soc.  A..  80,  466  (1908). 

f  Jour.  Am.  Chem.  Soc.,  38,  1907  (1916). 


186 


THEORETICAL  CHEMISTRY 


attached.  The  solution  to  be  studied  is  placed  inside  the  cylin- 
der and  around  the  clay  cell,  the  solvent  being  placed  inside  the 
latter.  It  will  be  seen  that  the  position  of  the  membrane  is 
similar  to  that  in  the  apparatus  of  Berkeley  and  Hartley. 

However,  instead  of  applying  mechanical  pressure  to  the  solu- 
tion until  equilibrium  was  established,  as  in  the  apparatus  of 
Berkeley  and  Hartley,  the  solution  was  allowed  to  develop  its  own 
pressure,  as  in  the  apparatus  of  Morse.  The  mercury  manometer 
of  the  Morse  apparatus  was  replaced  by  an  electrical  resistance 
gauge  in  which  use  is  made  of  the  principle  that  the  electrical 
resistance  of  certain  metallic  conductors  increases  very  nearly 
linearly  with  the  pressure.  By  means  of  this  gauge  it  was  possible 
to  measure  pressures  far  in  excess  of  those  which  the  mercury 
manometer  can  withstand.  The  following  table  gives  the  results 
of  measurements  of  the  osmotic  pressures  of  sucrose  solutions 
covering  the  complete  range  of  solubility. 

OSMOTIC  PRESSURE  OF  SUCROSE  SOLUTIONS  IN 
ATMOSPHERES 

(Temperature  30°) 


Cone.  gm.  per  1000  gm.  H2O 

Press.  (F.  &  M.) 

Press.  (B.  &  H.) 

202 

15.59 

15.48 

370 

29.78 

29.72 

569 

47.88 

48.81 

820 

73.06 

74.94 

1133 

109.10 

111.87 

1430 

148.80 

148.46 

The  values  given  in  the  last  column  are  calculated  from  the  data 
of  Berkeley  and  Hartley  for  sucrose  solutions  at  0°,  as  given  in  the 
preceding  table,  p.  183. 

Frazer  and  Lotz*  have  further  improved  the  apparatus  of 
Frazer  and  My  rick  by  substituting  the  water  interferometer,  as  a 
manometer,  in  place  of  the  electrical  resistance  gauge,  the  zero  of 
which  was  found  to  undergo  a  gradual  shift  due  to  hysteresis 
effects  in  the  resistance  coils.  The  possibility  of  employing  the 
water  interferometer  as  a  pressure  gauge,  is  based  upon  the 

*  Jour.  Am.  Chem.  Soc.,  43,  2501  (1921). 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE          187 

fact  that  when  pressure  is  applied  to  a  solution  in  one  of  the 
chambers  of  the  interferometer,  the  index  of  refraction  of  the 
solution  is  increased,  and  the  magnitude  of  the  applied  pressure 
can  be  measured  by  the  resultant  displacement  of  the  interfer- 
ence fringes.  With  this  apparatus,  osmotic  pressures  up  to  273 
atmospheres  have  been  measured,  while  the  time  required  for 
the  establishment  of  equilibrium  has  been  very  appreciably  re- 
duced. 

Lowering  of  Vapor  Pressure.  It  has  long  been  known  that 
the  vapor  pressure  of  a  solution  is  less  than  that  of  the  pure  sol- 
vent, provided  the  solute  is  non-  volatile.  The  investigations  of 
von  Babo  and  Wiillner  *  on  the  lowering  of  vapor  pressure  of 
various  liquids  when  non-volatile  substances  are  dissolved  in 
them,  resulted  in  the  following  generalizations:  —  (1)  The  lower- 
ing of  the  vapor  pressure  of  a  solution  is  proportional  to  the  amount 
of  solute  present;  and  (2)  For  the  same  solution,  the  lowering  of 
the  vapor  pressure  of  the  solvent  by  a  non-volatile  solute  is  at  all 
temperatures  a  constant  fraction  of  the  vapor  pressure  of  the  pure 
solvent. 

In  1887,  Raoult,f  as  the  result  of  an  exhaustive  experimental 
investigation,  enunciated  the  following  laws:  —  (1)  When  equi- 
molecular  quantities  of  different  non-volatile  solutes  are  dissolved  in 
equal  volumes  of  the  same  solvent,  the  vapor  pressure  of  the  solvent  is 
lowered  by  a  constant  amount;  and  (2)  The  ratio  of  the  observed 
lowering  of  the  vapor  pressure  to  the  vapor  pressure  of  the  pure  sol- 
vent is  equal  to  the  ratio  of  the  number  of  mots  of  solute  to  the  total 
number  of  mols  in  the  solution.  The  ratio  of  the  observed  lowering 
to  the  original  vapor  pressure  is  called  the  relative  lowering  of  the 
vapor  pressure.  Letting  pi  and  pz  denote  the  vapor  pressures  of 
solvent  and  solution,  Raoult's  second  law  may  be  formulated 
thus, 


"  N  +  n 

in  which  n  and  N  represent  the  number  of  mols  of  solute  and 
solvent  respectively.  Some  of  Raoult's  results  for  ethereal  solu- 
tions are  given  in  the  accompanying  table. 

*  Pogg.  Ann.,  103,  529  (1858). 

t  Compt.  rend.,  104,  1430  (1887);  Zeit.  phys.  Chem.,  2,  372  (1888);  Ann. 
Chem.  Phys.  (6),  15,  375  (1888). 


188 


THEORETICAL  CHEMISTRY 


LOWERING  OF  THE  VAPOR  PRESSURE  OF  ETHER 
PRODUCED   BY  VARIOUS  SOLUTES 


PI   -    P2 

PI   —   P2 

Solutes 

PI 

npi 

K 

Methyl  salicylate  
Methyl  benzoate  

152 
136 

2.91 
9.60 

0.026 
0.091 

0.0089 
0.0095 

0.71 
0.70 

Benzoic  acid  
Trichloracetic  acid  
Caprylic  alcohol  
Aniline                  

122 
163.5 
130 
93 

7.175 
11.41 
6.27 
7.66 

0.070 
0.120 
0.069 
0.081 

0.0097 
0.0105 
0.0110 
0.0106 

0.71 
0.71 
0.73 
0.71 

The  figures  in  the  last  column  of  the  table,  headed  K,  give  the 
lowering  of  the  vapor  pressure  produced  by  dissolving  one  mol  of 
solute  in  100  grams  of  solvent.  The  mean  value  of  pi  —  p^/npi 
for  14  different  solutes  studied  by  Raoult  was  0.0098;  this  is  in 
very  close  agreement  with  the  theoretical  value  of  the  relative  low- 
ering of  a  1  molar  per  cent  solution  calculated  as  follows :  — 

=  0.0099. 


pi 


100+1 


When  the  solution  is  very  dilute,  the  number  of  mols  of  solute 
is  negligible  in  comparison  with  the  number  of  mols  of  solvent, 
and  the  equation  of  Raoult  may  be  written 


Pi  ~ 


n 

N 


(5) 


Since  n  =  g/m,  and  N  =  W/M,  where  g  and  W  are  the  weights  of 
solute  and  solvent  respectively,  and  m  and  M  are  the  correspond- 
ing molecular  weights,  the  above  equation  becomes 


Pi  — 


PI 


_  gM 
Wm 


(6) 


This  equation  enables  us  to  calculate  the  molecular  weight  of  a 
dissolved  substance  from  the  relative  lowering  of  the  vapor 
pressure  produced  by  the  solution  of  a  known  weight  of  solute  in 
a  known  weight  of  solvent.  Solving  the  equation  for  m,  we  have 


m  = 


gM_        Pl 
W  '  p,  -  p2 


(7) 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE 


189 


an  illustration  of  the  application  of  this  equation,  we  may 
take  the  determination  of  the  molecular  weight  of  ethyl  benzoate 
from  the  following  experimental  data:  —  The  vapor  pressure  at 
80°  C.  of  a  solution  of  2.47  grams  of  ethyl  benzoate  in  100  grams 
of  benzene  is  742.6  mm.:  the  vapor  pressure  of  pure  benzene  at 
80°  C.  is  751.86  mm.  Substituting  in  the  equation,  we  have 
2.47  X  78  751.86 


m  = 


100 


751.86  -  742.6 


=  156. 


The  molecular  weight  calculated  from  the  formula, 
is  150. 

Connection  between  Lowering  of  Vapor  Pressure  and  Osmotic 
Pressure.  The  relation  between  osmotic  pressure  and  the  lower- 
ing of  vapor  pressure  has  been  derived  in 
the  following  manner  by  Arrhenius.*  Imag- 
ine a  very  dilute  solution  contained  in  the 
wide  glass  tube  A,  Fig.  64.  The  tube,  A, 
is  closed  at  its  lower  end  with  a  semi- 
permeable  membrane,  and  dips  into  a  ves- 
sel, B,  which  contains  the  pure  solvent.  The 
entire  apparatus  is  covered  by  a  bell-jar  C, 
and  the  enclosed  space  exhausted.  Let  h 
be  the  difference  in  level  between  the 
solvent  and  solution  when  equilibrium  is 
established,  that  is,  when  the  hydrostatic 
pressure  of  the  column  of  liquid  is  equal  to 
the  osmotic  pressure.  When  equilibrium  is 
attained,  the  vapor  pressure  of  the  solution 
at  the  height  "  h  "  will  be  equal  to  the  pressure  of  the  vapor 
of  the  solvent  at  this  height.  If  the  vapor  pressure  of  the  pure 
solvent  in  the  vessel  Bis  pi,  and  if  p2  is  the  vapor  pressure  of 
the  solution  at  the  height  h,  we  shall  have 

Pl-p,=  hd,  (8) 

where  d  denotes  the  density  of  the  vapor.     Let  v  be  the  volume  of 
1  mol  of  solvent  in  the  state  of  vapor,  then 

Plv  =  RT, 
and 

RT 

i)  =  — . 

Pi 
*  Zeit.  phys.  Chem.,  3,  115  (1889). 


1  

\ 
f 

i 

A 

/-: 

j 

rE=-j 

i 

™ 

B 

^. 

j 

~~~ 

Y 

irirE 

m 

Fig.  64 

190  THEORETICAL  CHEMISTRY 

If  the  molecular  weight  of  the  solvent  is  M,  we  may  replace  v 
by  M/d,  when  the  preceding  equation  becomes, 

M  =  RT 
d        pi 
or 

(9) 

The  solution  being  very  dilute  the  osmotic  pressure  may  be  cal- 
culated from  the  equation 

PV  =  nRT, 

where  P  is  the  osmotic  pressure  of  the  solution,  V  the  volume 
of  the  solution  containing  1  mol  of  solute,  and  n  the  number  of 
mols  of  solute  present.  If  s  represents  the  density  of  the  solvent 
and  also  of  the  solution,  (the  latter  being  very  dilute)  we  may 
write 

P  =  hs, 
and 


where  g  is  the  number  of  grams  of  the  solvent  in  which  the  n 
mols  of  solute  are  dissolved.  Substituting  these  values  of  P  and 
V  in  the  general  equation,  we  have 

PV  =  nRT  =  hg, 
and  solving  for  h,  we  have 


Substituting  the  values  of  d  and  h,  given  in  equations  (9)  and 
(10),  in  equation  (1),  we  have 

nRT    MPl      nMPl 
Pi-p2=—  -^=    — 

Rearranging  equation  (11),  and  remembering  that  N  =  g/M,  we 
obtain 


This  equation  it  will  be  seen,  is  identical  with  that  derived  experi- 
mentally by  Raoult  for  very  dilute  solutions. 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE  191 

By  an  application  of  thermodynamics  to  dilute  solutions, 
van't  Hoff  showed,  that  the  relation  between  osmotic  pressure  and 
the  relative  lowering  of  the  vapor  pressure  may  be  expressed  by 
the  equation, 

pi  -  p2  _  MP 
pi       ~'sRT1 

in  which  the  symbols  have  the  same  significance  as  above.     This 
equation  may  be  reconciled  easily  with  the  equation  of  Raoult. 
Thus,  if  n  in  equation  (12)  be  replaced  by  its  equal,  PV/RT, 
the  equation  becomes 

Pi  ~  P»  -     PV  (^ 

Pi       ~  NRT 

But  V  =  NM/s,  hence 

Pi-pt  =  MP 
pi       ~  sRT' 

This  equation  shows  that  the  relative  lowering  of  the  vapor  pressure 
is  directly  proportional  to  the  osmotic  pressure. 

The  Measurement  of  Vapor  Pressure.  The  difficulties  which 
attend  the  accurate  measurement  of  the  vapor  pressure  of  a  solu- 
tion by  the  static  method  have  already  been  mentioned.  While 
there  are  other  methods  which  are  preferable  for  the  determination 
of  the  molecular  weight  of  dissolved  substances,  the  vapor  pressure 
method  has  one  marked  advantage,  —  it  can  be  used  for  the  same 
solution  at  widely  divergent  temperatures.  The  method  devised 
by  Walker,  and  already  described  in  connection  with  the  deter- 
mination of  the  vapor  pressure  of  pure  liquids,  (p.  69)  is  well 
adapted  to  the  measurement  of  the  vapor  pressure  of  solutions. 

The  fundamental  importance  of  accurate  determinations  of 
the  lowering  of  the  vapor  pressure  of  various  solvents  by  dissolved 
substances  has  long  been  recognized.  With  this  end  in  view,  the 
inherent  defects  in  both  the  static  and  dynamic  methods  have  been 
made  the  subject  of  numerous  investigations,  and  during  the  past 
decade  improved  forms  of  apparatus  have  been  developed  which 
have  raised  these  methods  to  the  rank  of  methods  of  precision. 

The  outstanding  difficulties  in  all  static  methods  are,  (1)  the 
complete  removal  from  the  solution  of  dissolved  gases,  and  other 
constituents  more  volatile  than  the  solvent,  and  (2)  the  prevention 
of  inequalities  in  concentration  throughout  the  solution,  These 


192 


THEORETICAL  CHEMISTRY 


difficulties  have  been  almost  completely  overcome  by  Frazer 
and  Lovelace,*  who  have  devised  a  form  of  apparatus  in  which 
provision  is  made  for  the  removal  of  dissolved  air,  and  for  com- 
plete renewal  of  the  entire  surface  of  the  solution  by  vigorous 
stirring.  In  this  apparatus,  use  is  made  of  the  principle  of  the 
Rayleigh  manometer  f  to  measure  the  difference  between  the 
vapor  pressure  of  the  solution  and  that  of  the  pure  solvent.  The 


Fig.  65 

essential  features  of  this  extremely  sensitive  pressure  gauge  are 
shown  in  Fig.  65.  The  following  description  is  taken  from  the 
original  paper  of  Frazer  and  Lovelace.  "  The  two  glass  bulbs 
RRj  about  39  mm.  in  diameter,  are  'blown  on  a  glass  fork.  This 
is  connected  by  means  of  a  rubber  tube  with  a  mercury  reservoir 
which  may  be  adjusted  very  accurately  at  any  desired  height  by 
means  of  the  screw  /.  At  the  centers  of  the  bulbs  are  set  two 
glass  points.  The  side  limbs  PP,  communicate  with  the  systems, 

*  Jour.  Am.  Chem.  Soc.,  36,  2339  (1914). 
f  Trans.  Roy.  Soc.,  196,  205  (1901). 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE          193 

the  relative  pressures  in  which  are  to  be  measured.  The  bulbs 
are  set  in  plaster  in  an  iron  pot,  M,  which  is  mounted  on  an  axis, 
D,  at  right  angles  to  the  vertical  plane  passing  through  the  points. 
This  permits  of  rotation  by  means  of  the  screw  0. 

Now,  suppose  the  same  pressure  exists  in  the  two  bulbs.  By 
manipulation  of  the  two  screws  /  and  0,  the  one  serving  to  regulate 
the  height  of  the  mercury  and  the  other  to  rotate. the  manometer, 
the  two  points  can  readily  be  brought  into  coincidence  with  their 
images  in  the  mercury.  This  is  called  the  zero  position. 

Now,  suppose  the  pressure  in  one  bulb  to  be  slightly  increased. 
By  readjustment  of  the  mercury  level  and  again  rotating,  the 
points  can  a  second  time  be  brought  into  coincidence  with  their 
images.  It  is  obvious  that,  in  order  to  calculate  the  difference  in 
pressure  in  the  two  bulbs,  it  is  necessary  to  know  very  accurately 
the  distance  between  the  points  and  the  angle  of  rotation.  The 
angle  of  rotation  may  be  calculated  from  the  length  of  the  lever 
arm,  and  the  rotation  and  pitch  of  the  screw  0.  Or,  following 
Rayleigh,  we  may  determine  the  difference  in  pressure  by  means 
of  a  mirror,  telescope  and  scale,  and  this  is  the  method  actually 
used.  A  is  a  metallic  mirror  firmly  mounted  in  a  vertical  plane 
perpendicular  to  the  line  joining  the  two  glass  points.  The  axis 
of  rotation  lies  in  the  plane  of  the  mirror.  A  telescope,  with 
vertical  scale,  is  mounted  at  a  distance  of  a  little  over  three  meters 
in  front  of  the  mirror,  the  image  of  the  scale  being  at  all  times 
visible  through  the  telescope. 

If  d  represents  the  distance  between  the  points,  D  the  distance 
from  mirror  to  scale,  6  the  angle  of  rotation  from  the  zero  position, 
h  the  difference  in  pressure  in  the  two  limbs  of  the  manometer 
corresponding  to  this  angle  of  rotation  and  S  the  scale  deflection, 
then  the  following  equations  are  obtained: 

h  =  d  sin  0    and    S  =  D  tan  2  6 

and 

ds       sin  6 


h  = 


2  D     tan  2  d 


Now  for  all  values  of  6  up  to  1°,  j-^    ^^may  be  regarded  as  unity 

~2  tan  iL  \j 

and  our  formula  becomes,  h  =  ds/2  D.  For  the  particular  in- 
strument being  used  in  this  work  0=1°  corresponds  approxi- 
mately to  the  depression  of  a  3-molar  solution  of  a  non-electrolyte, 


194 


THEORETICAL  CHEMISTRY 


d  =  38.88  mm.  and  D  =  3350.6  mm.     Substituting  these  values 


and    making    S  =  1    we    get    h  = 


38.88  X  1 


=  0.00580.     This 


2  X  3350.6 

means  that  1  mm.  scale  deflection  from  zero  position  corresponds 
to  a  difference  in  pressure  in  the  two  bulbs  of  0.00580  mm.  By 
observing  the  points  through  microscopes  of  25  mm.  focus,  mounted 
on  the  instrument  and  rotating  with  it,  the  operator  can  set  the 
points  to  an  accuracy  of  0.1  mm.  on  the  scale,  which  corresponds 
to  a  difference  in  pressure  of  0.00058  mm.  This  is  the  limit  of 
accuracy  of  the  instrument  and  is  approximately  that  claimed 
by  Rayleigh  for  his  manometer,  of  which  the  one  used  in  this  work 
is  practically  a  reproduction." 

In  using  an  instrument  of  such  precision  the  authors  are  subject- 
ing the  static  method  of  measuring  vapor  pressures  to  the  severest 
test  that  could  be  applied.  The  results,  however,  have  fully  justi- 
fied its  use. 

In  the  following  table  are  given  the  results  of  a  series  of  meas- 
urements of  the  lowering  of  the  vapor  pressure  of  water  produced 
by  mannite. 

VAPOR  PRESSURE  OF  AQUEOUS  SOLUTIONS  OF  MANNITE  * 

(Temperature  20°) 


Vapor  pressure  lowering 

Lowering  per  mol 

Cone.  Wt.  Molal 

solute  (Obs.) 

(Obs.) 

(Calc.) 

0.0984 

0.0307 

0:0311 

0.3113 

0.1977 

0.0614 

0.0622 

0.3108 

0.2962 

0.0922 

0.0931 

0.3133 

0.3945 

0.1227 

0.1239 

0.3107 

0.4938 

0.1536 

0.1547 

0.3111 

0.5944 

0.1860 

0.1858 

0.3129 

0.6934 

0.2162 

0.2164 

0.3118 

0.7927 

0.2478 

0.2*69 

0.3126 

0.8922 

0.2792 

0.2775 

0.3129 

0.9908 

0.3096 

0.3076 

0.3124 

*  Frazer,  Lovelace  and  Rogers,  Jour.  Am.  Chem.  Soc.,  42,  1793  (1920). 

In  the  dynamic,  or  air-saturation  method,  of  determining  vapor 
pressures,  the  chief  sources  of  error  are,  (1)  in  the  measurement 
of  the  volume  of  aspirated  air,  and  (2)  in  maintaining  a  constant 
temperature  during  the  successive  determinations  of  the  vapor 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE 


195 


pressures  of  the  solvent  and  the  solution.  These  difficulties  may 
be  overcome  by  so  arranging  the  apparatus  that  the  same  quantity 
of  air  is  passed  over  the  solution  and  solvent  in  succession,  both 
liquids  being  contained  in  similar  "  saturators  "  which  are  im- 
mersed in  a  thermostat.  The  losses  in  weight  of  solution  and 
solvent,  due  to  the  removal  of  the  vapor  of  the  latter,  may  be 
determined  either  by  weighing  the  saturators  before  and  after 
an  experiment,  or  by  absorbing  the  moisture  which  is  present  in 
the  saturated  air  in  suitable  weighed  absorption  apparatus. 
The  former  method  has  been  followed  by  Berkeley,  Hartley  and 
Burton,*  while  Washburn  and  Heuse  f  have  adopted  the  latter. 
Both  methods  give  excellent  results.  The  following  table  contains 
the  results  of  Berkeley,  Hartley  and  Burton  for  solutions  of 
sucrose  at  0°. 

VAPOR  PRESSURE  OF  AQUEOUS  SOLUTIONS  OF  SUCROSE 

(Temperature  0°) 


Concentration  Gms. 

/o 

Osmotic  Press,  in 

Osmotic  Presa.  in 

Solute  to  100  Gms. 

atmos. 

atmos. 

Solvent 

ll 

(Calc.) 

(Obs.) 

56.50 

.03612 

43.91 

43.84 

81.20 

.05566 

67.43 

67.68 

112.00 

.08340 

100.53 

100.43 

141.00 

.11342 

134.86 

134.71 

183.00 

.  15908 

186.86 

217.50 

.19963 

230.70 

243.00 

.23060 

264.46 

The  second  column  of  the  table  gives  the  ratio  of  the  sum  of 
the  observed  losses  of  weight  of  the  solution  and  water  saturators, 
lo,  to  the  corresponding  loss  in  the  solution  saturators,  h.  The 
third  column  gives  the  values  of  the  osmotic  pressure  as  calcu- 
lated from  the  corresponding  vapor  pressure  lowerings,  while 
the  last  column  contains  the  observed  values  of  the  osmotic  pres- 
sure as  determined  by  Berkeley  and  Hartley.  J 

Elevation  of  the  Boiling-Point.  Just  as  the  vapor  pressure  of 
a  solution  is  less  than  that  of  the  pure  solvent,  so  the  boiling- 
point  of  a  solution  is  correspondingly  higher  than  the  boiling-point 

*  Phil.  Trans.  Roy.  Soc.,  A.  209,  177  (1909);  565,  295  (1919). 
t  Jour.  Am.  Chem.  Soc.,  37,  309  (1915). 
loc.  cit. 


196 


THEORETICAL  CHEMISTRY 


of  the  solvent.  It  follows  that  when  equimolecular  quantities 
of  different  substances  are  dissolved  in  equal  weights  of  the  same 
solvent,  the  elevation  of  the  boiling-point  is  constant.  Thus,  the 
molecular  weight  of  any  soluble  substance  may  be  determined  by 
comparing  its  effect  on  the  boiling-point  of  a  particular  solvent, 
with  that  of  a  solute  of  known  molecular  weight.  The  elevation 
in  boiling-point  produced  by  dissolving  1  mol  of  a  solute  in  1000 
grams,  of  a  solvent  is  termed  the  molar  elevation,  or  boiling-point 
constant  of  the  solvent.  In  determining  the  boiling-point  con- 
stant of  a  solvent,  a  fairly  dilute  solution  is  employed  and  the 
elevation  in  the  boiling-point  is  observed;  the  value  of  the  con- 
stant is  then  calculated  on  the  assumption  that  the  elevation  in 
boiling-point  is  proportional  to  the  concentration. 

If  g  grams  of  a  substance  of  unknown  molecular  weight  m, 
are  dissolved  in  G  grams  of  solvent,  and  the  boiling-point  is  raised 
dTb  degrees,  then,  since  m  grams  of  the  substance  when  dissolved 
in  1000  grams  of  solvent,  produce  an  elevation  of  K  degrees  \tlie 
molar  elevation),  it  follows  that 

1000  < 


therefore, 


m  =  1000  K 


GdTh 


(15) 


The  accompanying  table  gives  the  molar  boiling-point  constants 
for  some  of  the  more  common  solvents. 

MOLAR  BOILING-POINT  CONSTANTS  * 


Solvent 

Boiling-Point 
at  760  mm. 

dp/dT 
atmos./degree 

Kb 

Correction 
per  10  mm. 

Acetic  acid  
Benzene  
Carbon  disulphide  .... 
Chloroform 

118.51 
80.51 

46.00 
60  19 

0.0308 
0.0309 
0.0325 
0  0329 

3.28 
2.58 
2.40 
3  64 

0.08 
0.24 
0.20 
0  10 

Ethyl  alcohol  

78.26 

0.0379 

1.24 

0.10 

Ethyl  ether 

34.42 

0.0358 

2.21 

0  07 

Ethyl  acetate  
Methyl  alcohol  
Methyl  formate  
Methyl  acetate  
Propyl  alcohol  
Water  

77.13 
64.67 
31.92 
57.11 
97.14 
100.00 

0.0317 
0.0399 
0.0377 
0.0345 
0.0365 
0.0358 

2.90 
0.84 
1.63 
2.20 
1.71 
0.518 

0!l2 
0.09 
0.10 
0.09 
0.12 
0.07 

*  Rosanoff  and  Dunphy,  Jour.  Am.  Chem.  Soc.,  36,  1414,  (1914). 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE          197 

The  figures  given  in  the  last  column  are  to  be  used  in  correct- 
ing the  molar  boiling-point  constant  for  changes  in  barometric 
pressure:  the  corrections  are  to  be  added  to  Kb  for  every  10  mm. 
increase  in  pressure  over  760  mm.,  and  subtracted  for  every  10 
mm.  decrease  below  760  mm. 

Osmotic  Pressure  and  Boiling-Point  Elevation.  Imagine  a 
dilute  solution  containing  n  mols  of  solute  in  G  grams  of  solvent, 
and  let  dTb  be  the  elevation  in  the  boiling-point.  Suppose  a  large 
quantity  of  the  solution  to  be  introduced  into  a  cylinder,  fitted 
with  a  frictionless  piston,  and  closed  at  the  bottom  by  a  semi- 
permeable  membrane.  Let  the  cylinder  and  contents  be  raised 
to  the  absolute  temperature  T,  the  boiling-point  of  the  solvent, 
and  then  let  pressure  be  exerted  on  the  piston  just  sufficient  to 
overcome  the  osmotic  pressure  of  the  solution.  In  this  way,  let 
a  quantity  of  solvent  corresponding  to  1  mol  of  solute  be  forced 
through  the  semi-permeable  membrane.  The  volume  V,  thus 
expelled  is  the  volume  corresponding  to  G/n  grams  of  solvent. 
If  the  osmotic  pressure  of  the  solution  is  P,  then  the  work  done  in 
moving  the  piston  and  expelling  the  solvent  is  PV.  Now  let  the 
portion  of  the  solvent  which  has  been  forced  through  the  semi- 
permeable  membrane  be  vaporized.  For  this  operation,  G/n.Lv 
calories  will  be  required,  Lv  being  the  heat  of  vaporization  for  1 
gram  of  solvent  at  its  boiling-point.  Then  let  the  entire  system 
be  raised  to  the  boiling-point  of  the  solution,  (T  +  dTb),  the  pre- 
viously expelled  G/n  grams  of  vapor  being  allowed  to  mix  with 
the  solution.  The  heat  of  vaporization  lost  at  T,  is  thus  recov- 
ered at  the  slightly  higher  temperature,  (T  +  dTb).  Finally,  the 
entire  system  is  cooled  to  T,  and  is  thus  restored  to  its  original 
state. 

But  for  a  reversible  cyclical  process,  we  have 

HT 
dW  =  Q^,  (see  equation  (14)  p.  138) 

and  hence,  by  substitution,  we  obtain  the  expression 

PV    =  dTb 
G  T  ' 

n'Lv 
therefore, 

PVT  n 


198  THEORETICAL  CHEMISTRY 

But,  since  PV  =  RT,  equation  (16)  may  be  written, 

6     ~  T  '  /"7  " 

Lv      Or 

If  n  =  1  and  G  =  1000  grams,  then  dTb  =  Kb  (the  molar  elevation 
of  the  boiling-point),  or 

Kt 


~  1000  L/ 
Or  putting  R  =  2  calories,  we  have 

*,  =  °-M.        I  (17) 

Equation   (16)  shows  that  the  osmotic  pressure  of  a  solution  is 
directly  proportional  to  the  elevation  of  the  boiling-point. 

Equation  (17)  was  originally  derived  by  van't  Hoff  at  about  the 
time  when  Raoult  determined  the  values  of  Kb  experimentally. 
The  calculated  values  of  Kb  are  in  close  agreement  with  the 
values  obtained  experimentally  by  Raoult  and  others.  As  an 
example,  the  calculated  value  of  the  molecular  elevation  for- 
water,  the  heat  of  vaporization  of  which  at  100°  C.  is  537  calories,  is 

v        0.002  X  (373)2 


a  value  which  is  in  exact  agreement  with  that  given  in  the  table. 
Experimental  Determination  of  Boiling-Points.  One  of  the 
simplest  and  most  convenient  of  the  various  forms  of  apparatus 
which  have  been  devised  for  the  determination  of  the  boiling-points 
of  solutions,  is  that  developed  by  Mathews  *  from  an  earlier  form 
of  apparatus  described  by  Bigelow.  f  In  this  apparatus,  the  heat 
is  supplied  electrically,  this  having  been  found  to  be  far  superior 
to  gas  heating  in  which  protecting  mantles,  glass  beads  and  various 
other  troublesome  accessories  are  necessary.  A  diagram  of  the 
apparatus  is  shown  in  Fig.  66.  It  will  be  seen  to  consist  of  a 
boiling  vessel  A,  closed  by  a  cork  stopper,  through  which  pass  a 
Beckmann  thermometer,  the  return  tube  from  the  condenser  and 
the  two  small  glass  tubes  B  containing  the  wires  through  which 
the  current  is  led  to  the  platinum  heating  coil.  The  boiling  vessel 

*  Trans.  Am.  Electrochem.  Soe.  19,  81  (1911), 
f  Am.  Chem.  Jour.,  22,  280  (1899). 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE          199 


is  entirely  enveloped  within  an  un-silvered  Dewar  cylinder  C. 
Circulation  of  air  between  the  boiling  vessel  and  the  vacuum 
jacket  is  prevented  by  means  of  a  rubber  gasket,  or  by  cotton  pack- 
ing.    The  bulb  of  the  thermometer  is  placed  in  the  center  of  the 
large  vertical  coil,  as  shown  in  the  diagram.     Bubbles  of  vapor 
in  rising  do  not  touch  the  bulb  of  the  thermometer,  but  rise  directly 
to  the  surface  of  the  liquid.     The  vortex  motion  produced  by  the 
ascending  stream  of  bubbles 
serves  to  stir  the  liquid  very 
effectively.       The    flattened 
bottom  of  the  boiling  vessel 
makes  it  possible   to    bring 
the  heating  coil  so  close  to 
the  lower  end  of  the  vessel 
that  inequality  of   tempera- 
ture, due  to  a  more  or  less 
stagnant  layer  of  liquid,  is 
avoided.      The  heating   ele- 
ments consist  of  two  strands 
of  platinum  wire   0.38  mm. 
in    diameter  and   about   20 
cm.    in    length.     These    are 
twisted  together  and  a  cur- 
rent of  from  10  to   14  am- 
peres at  110  volts  is  used. 
With     a     current     of     this 
strength  there  is  far  less  superheating  than  when  a  coil  of  finer 
wire  and  greater  length  is  used  with  a  current  of  lower  amperage. 
The  constancy  of  the  current  is  controlled  by  means  of  a  suitable 
rheostat   with   a   sliding   contact.     Inasmuch   as    constancy   of 
boiling-point  is  largely  dependent  upon  the  constancy  of  the  rate 
at  which  the  condensed  and  cooled  liquid  returns  from  the  con- 
denser, it  is  very  necessary  that  heat  be  supplied  at  a  perfectly 
fixed   rate.     Naturally   such   accurate   control   is   all   the   more 
necessary  when  large  currents  are  sent  through  small  resistances, 
but  this  fs  a  requirement  easily  satisfied,  and  constitutes  no  objec- 
tion to  the  use  of  the  heavier  currents.     The  condensed  liquid 
drops  freely  from  the  condenser  to  the  violently  agitated  boil- 
ing liquid  below.     Fluctuations  in  temperature  always  result  if 
the  liquid  runs  down  the  thermometer,  the  outside  walls  or  the 


Fig.   66 


200 


THEORETICAL  CHEMISTRY 


glass  tubes,  these  fluctuations  being  due  to  imperfect  mixing. 
A  known  weight  of  liquid  in  A  is  boiled  until  the  thermometer 
remains  constant;  this  temperature  is  taken  as  the  boiling-point 
of  the  liquid.  The  apparatus  is  now  emptied  and  dried.  A 
weighed  amount  of  solute  is  added  and  the  boiling-point  of  the 
solution  is  determined.  By  means  of  a  capillary  siphon  a  sample 
of  the  solution  may  be  removed  for  analysis.  The  difference 
between  the  readings  of  the  thermometer  when  immersed  in 
the  solution,  and  in  the  solvent  alone,  gives  the  boiling-point 
elevation. 

In  determining  the  boiling-point  of  a  pure  liquid  it  is  customary 
to  place  the  bulb  of  the  thermometer  in  the  vapor  of  the  boilmg 
liquid.  Under  these  conditions,  the  thermometer  bulb  becomes 
coated  with  a  thin  film  of  the  liquid  in  contact  with  its  vapor,  and, 
by  thus  avoiding  superheating,  an  extremely  accurate  determina- 
tion of  the  boiling-point  of  the  liquid  is  insured.  In  determining 
the  boiling-point  of  a  solution,  however,  where  the  thermometer 
bulb  is  immersed  in  the  liquid,  there  is  always  an  appreciable  error 

due  to  superheating,  and 
at  best,  the  observed  boil- 
ing-point is  to  be  regarded 
as  only  an  approximation 
to  the  true  boiling-point 
of  the  solution. 

An  ingenious  form  of 
boiling-point  apparatus  has 
been  devised  by  Cottrell* 
by  means  of  which  it  is 
possible  to  secure  a  perfect 
equilibrium  between  the 
liquid  and  vapor  phases  of 
a  solution,  and  to  deter- 
mine with  a  high  degree 
of  accuracy  the  tempera- 
ture of  this  equilibrium. 

The  novel  feature  of  this  form  of  boiling-point  apparatus  consists 
in  placing  the  thermometer  bulb  in  the  vapor  phase,  as  in  the 
case  of  a  pure  liquid,  and  causing  the  boiling  liquid  to  pump 
some  of  itself  over  the  thermometer  bulb  in  a  thin  film,  thus 
*  Jour.  Am.  Chem.  Soc.,  41,  721  (1919). 


Fig.  67 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE          201 

reproducing,  as  far  as  possible,  the  same  relations  of  vapor 
and  liquid  to  the  thermometer  as  exist  in  determinations  of 
the  boiling-point  of  a  pure  liquid. 

The  apparatus  is  shown  diagrammatically  in  Fig.  67.  The  tube 
D  functions  as  a  pump,  and  insures  the  passage  of  an  intimate  mix- 
ture of  solution  and  vapor  over  the  thermometer  bulb.  A  glass 
sheath,  /,  is  attached  to  the  ground  glass  stopper  and  serves  to 
protect  the  thermometer  bulb  from  cooling  the  influence  of  the 
condensed  solvent  as  it  flows  back  into  the  boiling  vessel.  The  so- 
lute is  introduced  through  the  condenser  tube  H.  When  the  boil- 
ing-point of  a  solution  becomes  constant  to  within  0.001°,  the  source 
of  heat  is  momentarily  removed  and  the  capillary  U-tube,  B,  is 
surrounded  by  a  bath  of  ice  water.  By  applying  air  pressure  at 
the  open  end  of  the  condenser  tube,  H,  a  sample  of  the  solution 
can  be  removed  for  analysis.  In  this  way  the  exact  concentration 
of  the  liquid  phase  can  be  obtained. 

The  following  table  contains  the  results  of  an  investigation, 
on  the  boiling-points  of  solutions  of  naphthalene  in  benzene 
carried  out  with  the  Cottrell  apparatus  by  Washburn  and  Read.  * 

BOILING-POINTS  OF  SOLUTIONS  OF  NAPHTHALENE 
IN  BENZENE 


dTb(obs.) 

xi  (obs.) 

xi  (calc.) 

1.478 

0.04169 

0.04293 

2.430 

0.07012 

0.06938 

3.890 

0.10779 

0.10820 

4.967 

0.13529 

0.13553 

5.238 

0.14220 

0.14236 

6.368 

0.17204 

0.16947 

The  figures  in  the  first  column  give  the  values  of  the  measured 
elevations  of  the  boiling-point,  while  those  of  the  second  column 
give  the  corresponding  concentrations  expressed  as  mol  fractions 
of  solute.  The  values  of  the  concentrations,  calculated  on  the 
assumption  that  the  solutions  obey  Raoult's  law  are  given  in  the 
last  column. 

Lowering  of  the  Freezing-Point.  Of  all  the  methods  employed 
for  the  determination  of  molecular  weights  in  solution,  the  freez- 
ing-point method  is  the  most  accurate  and  the  most  widely  used. 
*  Jour.  Am.  Chem.  Soc.,  41,  734  (1919). 


202 


THEORETICAL  CHEMISTRY 


It  was  pointed  out  by  Blagden  *  over  a  century  ago,  that  the  de- 
pression of  the  freezing-point  of  a  solvent  by  a  dissolved  substance  is 
directly  proportional  to  the  concentration  of  the  solution.  When 
equimolecular  quantities  of  different  substances  are  dissolved  in 
equal  volumes  of  the  same  solvent,  the  lowering  of  the  freezing- 
point  is  constant.  The  molecular  weight  of  any  soluble  sub- 
stance can  be  found,  as  in  the  boiling-point  method,  by  comparing 
its  effect  on  the  freezing-point  of  a  solvent  with  that  of  a  solute 
of  known  molecular  weight.  The  molar  lowering  of  the  freezing- 
point,  or  the  freezing-point  constant,  of  a  solvent  is  defined  as  the 
depression  of  the  freezing-point  produced  by  dissolving  1  mol  of 
solute  in  1000  grams.  The  freezing-point  constants  of  a  few  com- 
mon solvents  are  given  in  the  following  table :  — 

MOLAR  FREEZING-POINT  CONSTANTS 


Solvent 

Freezing-Point 

Molar  Depression,  Kf. 

Acetic  acid  
Benzene  

17 
5.5 

3.9 
5.12 

Benzil  
Bromine  
Ethylene  bromide  
Formic  acid 

94 
-7.3 
10 

8 

10.5 
9.7 
12.5 

2.8 

Naphthalene  
.Nitrobenzene  
Phenol  

80 
5.3 

38.5 

6.8 
7.0 
7.4 

Tribromphenol  
Urethane  

95 
49 

20.4 
5.14 

Water 

0 

1  855 

p-Xylol 

16 

4.3 

A  formula  analogous  to  that  employed  for  the  calculation  of 
the  molecular  weight  of  a  dissolved  substance  from  the  elevation  it 
produces  in  the  boiling-point  of  a  solvent,  may  be  used  for  the 
calculation  of  molecular  weight  from  freezing-point  depression. 
Thus,  if  g  grams  of  solute,  when  dissolved  in  G  grams  of  solvent, 
produce  a  depression  d  T/  in  the  freezing-point  of  the  solvent,  the 
molecular  weight  m}  is  given  by  the  formula, 


ra  =  1000  Kf 


where  K/  is  the  molecular  lowering  of  the  freezing-point. 
*  Phil.  Trans.,  78,  277  (1788). 


(18) 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE          203 

Osmotic  Pressure  and  Freezing-Point  Depression.     Let  dTf 

be  the  freezing-point  depression  produced  by  n  mols  of  solute  in 
G  grams  of  solvent,  the  solution  being  dilute.  Imagine  a  large 
quantity  of  this  solution  to  be  confined  within  a  cylinder  fitted 
with  a  frictionless  piston,  and  the  bottom  of  the  cylinder  to  be 
closed  by  a  semi-permeable  membrane.  Let  the  cylinder  and  con- 
tents be  cooled  to  the  freezing  temperature  of  the  solvent  T,  and 
then  let  pressure  be  applied  to  the  piston  until  a  quantity  of  solvent 
corresponding  to  1  mol  of  solute  is  forced  through  the  semi-perme- 
able membrane.  This  requires  an  expenditure  of  energy  equiva- 
lent to  P  V,  where  P  is  the  osmotic  pressure  of  the  solution  and  V 
is  the  volume  of  solvent  expelled.  The  volume  V  is  clearly  the 
volume  of  G/n  grams  of  solvent.  Now  let  the  expelled  portion 

ri 

of  solvent  be  frozen,  and  the  system  thereby  deprived  of  -  L/ 

Ti 

calories  of  heat,  where  L/  is  the  heat  of  fusion  of  1  gram  of 
the  solvent  at  the  temperature  T.  The  temperature  of  the 
solution  is  then  lowered  to  its  freezing-point  (T  —  dTf),  and 
the  G/n  grams  of  solidified  solvent  dropped  into  it.  The  solid- 
ified solvent  melts,  thereby  restoring  to  the  system  at  the  tem- 
perature (T  —  dTf),  the  heat  of  fusion  formerly  taken  from  it. 
Finally,  the  temperature  of  the  system  is  raised  to  T,  the  initial 
temperature  of  the  cycle.  Applying  the  familiar  thermodynamic 
relation, 

dW  =  Q™  (see  equation  (14)  p.  138), 


we  have 

P17  /7T. 

(19) 


PV      dTf 


G  ,         T 
nL' 


From  which  we  obtain 

PVT  n 
dTf=   ~Lf~'G' 

But  PV  =  RT,  hence  equation  (19)  becomes 


204 


THEORETICAL  CHEMISTRY 


If  n  =  1  and  G  =  1000  grams,  then  dTf  =  Kf,  the  molar  lowering 
of  the  freezing-point,  and 

RT* 
1000  L/ 


' 


Or  putting  R  =  2  calories,  we  have 


(20) 


It  is  evident  from  equation  (19)  that  the  osmotic  pressure  of  a 
solution  is  directly  proportional  to  the  freezing-point  depression. 

The  agreement  between  the  observed  and  the  calculated  values 
of  Kf  is  very  satisfactory,  as  the  following  calculation  for  water 

shows:  — 

0.002  X  (273)2 


Kf 


80 


=  1.86. 


It  is  of  interest  to  note  that  the  calculated  value  of  Kf  for  water  is 
lower  than  the  experimental  values  originally 
obtained  by  Raoult  and  others.  Subsequent 
experiments,  carried  out  with  greater  care  and 
better  apparatus,  by  Raoult,  Abegg  and  Loomis 
gave  values  in  close  agreement  with  that 
derived  theoretically.  .0.- 

Experimental  Determination  of  Freezing- 
Points.  The  apparatus  almost  universally 
employed  for  the  determination  of  molecular 
weights  by  the  freezing-point  method  is  that 
devised  by  Beckmann,*  and  shown  in  Fig.  68. 
It  consists  of  a  thick- walled  test  tube  A,  pro- 
vided with  a  side  tube,  and  fitted  into  a  wider 
tube  A  i.  The  whole  is  fitted  into  the  metal 
cover  of  a  large  battery  jar,  which  is  filled 
with  a  freezing  mixture  whose  temperature  is 
several  degrees  below  the  freezing-point  of  the 
solvent.  The  tube  A  is  closed  by  a  cork 
stopper,  through  which  passes  the  thermometer 
and  stirrer.  The  thermometer  is  generally  of 
the  Beckmann  differential  type. 


Fig.  68 


In  carrying  out  a  determination  with  the  Beckmann  apparatus, 
*  Zeit.  phys.  Chem.,  2,  683  (1888). 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE         205 

a  weighed  quantity  of  solvent  is  placed  in  A,  and  the  tempera- 
ture of  the  refrigerating  mixture  is  regulated  so  as  to  be  not 
more  than  5°  below  the  freezing-point  of  the  solvent.  The  tube 
A  is  removed  from  its  jacket,  and  is  immersed  in  the  freezing 
mixture  until  the  solvent  begins  to  freeze.  It  is  then  replaced  in 
the  jacket  AI,  and  the  solvent  is  vigorously  stirred.  The  tem- 
perature rises  during  the  stirring  until  the  true  freezing-point 
is  reached,  after  which  it  remains  constant.  This  temperature 
is  taken  as  the  freezing-point  of  the  solvent.  The  tube  A  is 
now  removed  from  the  freezing  mixture,  and  a  weighed  amount 
of  the  substance  whose  molecular  weight  is  to  be  determined 
is  introduced.  When  the  substance  has  dissolved,  the  tube  is 
replaced  in  A\  and  the  solution  cooled  not  more  than  a  degree 
below  its  freezing-point.  A  small  fragment  of  the  solid  solvent 
is  dropped  into  the  solution,  which  is  then  stirred  vigorously 
until  the  thermometer  remains  constant.  The  maximum  tem- 
perature is  taken  as  the  freezing-point  of  the  solution.  The 
difference  between  the  freezing-points  of  solution  and  solvent  is 
the  depression  sought. 

In  order  to  obtain  trustworthy  results  with  the  freezing-point 
method,  it  is  necessary  that  only  the  pure  solvent  separate  out 
when  the  solution  freezes,  and  that  excessive  overcooling  be 
avoided.  When  too  great  overcooling  occurs,  the  subsequent 
freezing  of  the  solution  results  in  the  separation  of  so  large  an 
amount  of  solvent  in  the  solid  state,  that  the  observed  freezing- 
point  corresponds  to  the  equilibrium  temperature  of  a  more 
concentrated  solution  than  that  originally  prepared. 

If  the  overcooling  of  the  solution  in  degrees  be  represented  by 
u,  the  heat  of  fusion  of  1  gram  of  solvent  at  the  freezing-point  by 
Lfj  and  the  specific  heat  of  the  solvent  by  c,  then  the  fraction  of 
the  solvent  which  will  solidify,  /,  may  be  calculated  by  the  formula, 

/  =  g-  (21) 

When  water  is  used  as  the  solvent,  c  =  I  and  Lf  =  80.  There- 
fore, for  every  degree  of  overcooling,  the  fraction  of  the  solvent 
separating  as  ice  will  be  1/80,  and  the  concentration  of  the  original 
solution  is  increased  by  just  so  much.  It  is  simpler,  however,  to 
apply  the  correction  directly  to  the  freezing-point  depression 
instead  of  to  the  concentration. 


206 


THEORETICAL  CHEMISTRY 


S- 


The  foregoing  method  for  the  measurement  of  freezing-points  is 
sufficiently  accurate  for  all  ordinary  laboratory  requirements, 
where  a  precision  not  greater  than  0.001°  in  the  measurement  of 
temperature  is  required.  For  precise  measurements,  however, 
the  following  conditions  are  essential:  —  (1)  Rapid  and  certain 
attainment  of  the  equilibrium  temperature;  (2)  accurate  meas- 
urement of  the  equilibrium  temperature,  preferably  by  means  of 

a  differential  method 
in  which  the  freezing- 
point  depression  is 
measured  directly;  and 
(3)  exact  determina- 
tion of  the  concentra- 
tion of  the  solution 
when  equilibrium  is 
attained.  Numerous 
forms  of  apparatus 
have  been  designed 
with  a  view  to  meeting 
these  conditions,  but 
none  is  superior  to  that 
devised  by  Adams.* 
In  this  form  of  ap- 
paratus the  above  re- 
quirements were  satis- 
fied as  follows:  —  (1) 
By  thoroughly  mixing 
the  solution  with  a 
large  amount  of  finely 
crushed  ice  by  means 
of  an  efficient  stirrer; 
(2)  by  measuring  direct- 
ly the  freezing-point 
69  depression  by  means 

of   a   multiple-junction 

thermolement  and  sensitive  potentiometer;    and  (3)  by  analyz- 
ing  the    equilibrium   solution  by  means  of   special  optical  ap- 
paratus with  which  the  concentration  could  be  determined  with 
an  accuracy  of  two  parts  of  solute  in  a  million  of  solvent. 
*  Jour.  Am.  Chem.  Soc.,  37,  481  (1915). 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE 


207 


A  diagram  of  the  apparatus  is  shown  in  Fig.  69.  Two  similar 
vacuum-jacketed  cylinders,  A  and  B,  are  completely  surrounded 
with  crushed  ice  contained  in  a  large  felt-covered  tank.  The 
cylinder  A  contains  about  400  grams  of  ice  and  600  grams  of 
water,  while  the  cylinder  B  contains  the  same  respective  amounts 
of  ice  and  solution.  The  liquid  in  each  cylinder  is  thoroughly 
mixed  by  means  of  the  stirrers,  S,S,  which,  when  moved  by 
the  glass  rods,  R,R,  pump  liquid  from  the  bottom  of  each  jar 
and  discharge  it  over  the  ice.  The  difference  in  temperature  be- 
tween the  contents  of  A  and  B  is  measured  directly  by  means 
of  the  thermoelement  T.  When  equilibrium  is  attained,  a  sample 
for  analysis  is  withdrawn  by  means  of  P,  which  also  serves  to 
introduce  both  water  and  solution  into  B. 

The  following  table  gives  the  results  of  a  series  of  determina- 
tions of  the  freezing-points  of  solutions  of  mannite  made  with 
the  Adams  apparatus. 

FREEZING-POINTS  OF  AQUEOUS  SOLUTIONS  OF  MANNITE* 


Concentration   Milli- 
mols  per  1000  gm.  H2O. 
NX  103 

Freezing-Point  Depression 

N  X  10'  (calc.) 

dTf  (obs.) 

dTf  (calc.) 

4.02 
8.42 
14.04 
28.29 
62.59 

0.0075 
0.0157 
0.0260 
0.0525 
0.1162 

0.0075 
0.0156 
0.0261 
0.0525 
0.1162 

4.04 
8.45 
13.99 
28.26  ' 
62.58 

*  Adams,  loc.  cit. 

The  third  column  contains  the  calculated  values  of  the  freezing- 
point  depression,  while  the  last  column  gives  the  concentrations 
calculated  from  the  observed  freezing-point  depressions.  It 
will  be  seen  that  the  solution  follows  the  freezing-point  law  very 
closely,  i.e.,  the  depression  of  the  freezing-point  is  directly  pro- 
portional to  the  concentration. 

Molecular  Weight  in  Solution.  As  has  been  pointed  out,  the 
molecular  weight  of  a  dissolved  substance  can  be  readily  calcu- 
lated, provided  that  the  osmotic  pressure  of  a  dilute  solution  of 
known  concentration  at  known  temperature  is  determined.  But 
the  experimental  difficulties  attending  the  direct  measurement 


208  THEORETICAL  CHEMISTRY 

of  osmotic  pressure  are  so  great,  that  it  is  customary  to  employ 
other  methods  based  upon  properties  of  dilute  solutions  which 
are  proportional  to  osmotic  pressure.  We  have  shown  that  in 
dilute  solutions  osmotic  pressure  is  directly  proportional,  (1)  to 
the  relative  lowering  of  the  vapor  pressure,  (2)  to  the  elevation 
of  the  boiling-point,  and  (3)  to  the  depression  of  the  freezing-point. 

From  this  it  follows,  that  equimolecular  quantities  of  different 
substances  dissolved  in  equal  volumes  of  the  same  solvent,  exert  the 
same  osmotic  pressure,  and  produce  the  same  relative  lowering  of 
vapor  pressure,  the  same  elevation  of  boiling-point,  and  the  same 
depression  of  freezing-point.  Since  equimolecular  quantities  of 
different  substances  contain  the  same  number  of  molecules,  it  is 
evident  that  the  magnitude  of  osmotic  pressure,  relative  lowering  of 
vapor  pressure,  elevation  of  boiling-point  and  depression  of  freezing- 
point,  is  dependent  upon  the  number  of  dissolved  units  present  in  the 
solution  and  is  independent  of  their  nature. 

It  has  been  pointed  out  by  Nernst  that  any  process  which 
involves  the  separation  of  solvent  from  solute,  may  be  employed 
for  the  determination  of  molecular  weights.  A  little  reflection 
will  convince  the  reader  that  the  four  methods  discussed  in  this 
chapter  involve  such  separation.  Both  van't  Hoff  and  Raoult 
emphasized  the  fact  that  the  formulas  derived  for  the  deter- 
mination of  molecular  weights  in  solution  depend  upon  assump- 
tions which  are  valid  only  for  dilute  solutions.  It  follows,  there- 
fore, that  we  are  not  justified  in  applying  these  formulas  to  con- 
centrated solutions.  Up  to  the  present  time  we  have  no  satis- 
factory theory  of  concentrated  solutions,  neither  can  we  state 
up  to  what  concentration  the  gas  laws  apply. 

REFERENCES 

Osmotic  Pressure.     Alexander  Findlay. 

Osmotic  Pressure.     (A  general  discussion  held  by  the  Faraday  Society,  May 

1,  1917.)     Trans.  Faraday  Soc.,  13,  (1917). 
The  Osmotic  Pressure  of  Aqueous  Solutions.     Morse.     Carnegie  Inst.  Publ. 

198. 

PROBLEMS 

1.  At  10°  C.  the  osmotic  pressure  of  a  solution  of  urea  is  500  mm.  of 
mercury.  If  the  solution  is  diluted  to  ten  times  its  orginal  volume, 
what  is  the  osmotic  pressure  at  15°  C.?  Ans.  50.89  mm. 

V     2.  The  osmotic  pressure  of  a  solution  of  0.184  gram  of  urea  in  100  cc. 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE          209 

of  water  was  56  cm.  of  mercury  at  30°  C.    Calculate  the  molecular  weight 
of  urea.  Ans.  62.12. 

3.  At  24°  C.  the  osmotic  pressure  of  a  cane  sugar  solution  is  2.51  atmos- 
pheres.    What  is  the  concentration  of  the  solution  in  mols  per  liter? 

4.  At  25°.  1  C.  the  osmotic  pressure  of  solution  of  glucose  containing 
18  grams  per  liter  was  2.43  atmospheres.     Calculate  the  numerical  value 
of  the  constant  R,  when  the  unit  of  energy  is  the  gram-centimeter. 

5.  The  vapor  pressure  of  ether  at  20°  C.  is  442  mm.  and  that  of  a  solu- 
tion of  6.1  grams  of  benzoic  acid  in  50  grams  of  ether  is  410  mm.  at  the 
same  temperature.     Calculate  the  molecular  weight  of  benzoic  acid  in 
ether.  Ans.  124. 

6.  At  10°  C.  the  vapor  pressure  of  ether,  is  291.8  mm.  and  that  of  a 
solution  containing  5.3  grams  of  benzaldenyde  in  50  grams  of  ether  is 
271.8  mm.     What  is  the  molecular  weight  of  benzaldehyde? 

7.  A  solution  containing  0.5042  gram  of  a  substance  dissolved  in  42.02 
grams  of  benzene  boils  at  80°.  175  C.     Find  the  molecular  weight  of  the 
solute,  having  given  that  the  boiling-point  of  benzene  is  80°.00  C.,  and  its 
heat  of  vaporization  is  94  calories  per  gram.  Ans.  181.9. 

8.  A  solution  containing  0.7269  gram  of  camphor  (mol.  wt.  =  152) 
in  32.08  grams  of  acetone  (boiling-point  =  56°.30  C.)  boiled  at  56°.55  C. 
What  is  the  molar  elevation  of  the  boiling-point  of  acetone?     What  is 
its  heat  of  vaporization?      Ans.  KI,  =  1.67;  Lv  =  129.5  cals.  per  gram. 

9.  A  solution  of  9.472  grams  of  CdI2  in  44.69  grams  of  water  boiled  at 
100°. 303  C.      The  heat  of  vaporization  of  water  is  536  calories  per  gram. 
What  is  the  molecular  weight  of  CdI2  in  the  solution?     What  conclusion 
as  to  the  state  of  CdI2  in  solution  may  be  drawn  from  the  result? 

10.  The  freezing-point  of  pure  benzene  is  5°. 440  C.  and  that  of  a  solu- 
tion containing  2.093  grams  of  benzaldehyde  in  100  grams  of  benzene  is 
4°. 440  C.     Calculate  the  molecular  weight  of  benzaldehyde  in  the  solu- 
tion.   Kf  for  benzene  is  5.12.  Ans.  104.6. 

11.  A  solution  of  0.502  gram  of  acetone  in  100  grams  of  glacial  acetic 
acid  gave  a  depression  of  the  freezing-point  of  0°.339  C.     Calculate  the 
molar  depression  for  glacial  acetic  acid.  Ans.  3.9. 

12.  By  dissolving  0.0821  gram  of  m-hydroxybenzaldehyde  (C7H6O2)  in 
20  grams  of  naphthalene  (melting-point  80°.  1  C.)  the  freezing-point  is 
lowered  by  0°.232  C.     Assuming  that  the  molecular  weight  of  the  solute 
is  normal  in  the  solution,  calculate  the  molar  depression  for  naphthalene 
and  the  heat  of  fusion  per  gram. 

Ans.  Kf  =  6.90;  Lf  =  36.2  cals.  per  gram. 


CHAPTER   IX 
SOLUTIONS   OF  ELECTROLYTES 

Abnormal  Solutes.  As  has  already  been  pointed  out,  the  ac- 
ceptance of  Avogadro's  hypothesis  was  greatly  retarded  by  the 
discovery  of  certain  substances  whose  vapor  densities  were  ab- 
normal. Thus,  the  vapor  density  of  ammonium  chloride  is  ap- 
proximately one-half  of  that  required  by  the  formula  NH4C1,  while 
the  vapor  density  of  acetic  acid  corresponds  to  a  formula  whose 
molecular  weight  is  greater  than  that  calculated  from  the  formula, 
C2H402.  The  anomalous  behavior  of  ammonium  chloride  and 
kindred  substances  has  been  shown  to  be  due,  not  to  a  failure  of 
Avogadro's  law,  but  to  a  breaking  down  of  the  molecules  —  a 
process  known  as  dissociation.  The  abnormally  large  molecular 
weight  of  acetic  acid  on  the  other  hand,  has  been  ascribed  to  a 
process  of  aggregation  of  the  normal  molecules,  known  as  asso- 
ciation. In  extending  the  gas  laws  to  dilute  solutions  similar 
phenomena  have  been  encountered. 

Association  in  Solution.  When  the  molecular  weight  of  acetic 
acid  in  benzene  is  determined  by  the  freezing-point  method,  the 
depression  of  the  freezing-point  is  abnormally  small  and  conse- 
quently, the  molecular  weight  will  be  greater  than  that  correspond- 
ing to  the  formula,  C2H4O2.  Acetic  acid  in  benzene  solution  is 
thus  shown  to  be  associated.  Almost  all  compounds  containing 
the  hydroxyl  and  cyanogen  groups  when  dissolved  in  benzene  are 
found  to  be  associated.  Solvents,  such  as  benzene  and  chloro- 
form, are  frequently  termed  associating  solvents,  although  it  is 
doubtful  whether  they  exert  any  positive  associating  action. 
There  is  considerable  experimental  evidence  to  show  that  those 
substances  whose  molecules  are  associated  in  beneene  and  chloro- 
form solution,  are  also  associated  in  the  free  condition.  Just  as 
the  depression  of  the  freezing-point  of  a  solution  of  an  associated 
substance  is  abnormally  small,  so  its  osmotic  pressure  and  other 
related  properties  will  be  less  than  the  calculated  values. 

210 


SOLUTIONS  OF  ELECTROLYTES  211 

Dissociation  in  Solution.  Van't  Hoff*  pointed  out  that  the 
osmotic  pressures  of  solutions  of  most  salts,  of  all  strong  acids,  and 
of  all  strong  bases  are  much  greater  for  all  concentrations  than 
would  be  expected  from  the  osmotic  pressures  of  solutions  of  sub- 
stances, like  cane  sugar  or  urea,  for  corresponding  concentrations. 
Although  van't  Hoff  was  unable  to  offer  any  satisfactory  explana- 
tion of  this  anomaly,  he  showed  that  the  fundamental  equation, 
PV  =  RT,  would  still  hold,  provided  an  empirical  factor,  i,  were 
introduced  and  the  equation  written  in  the  form, 

PV  =  iRT.  (1) 

It  was  originally  believed  that  the  factor,  i,  was  a  constant,  but 
subsequently,  when  Arrhenius  had  advanced  his  theory  of  electro- 
lytic dissociation,  it  became  apparent  that  i  must  increase  as  the 
concentration  diminishes.  If  the  osmotic  pressure  of  some  sub- 
stance, like  cane  sugar,  which  behaves  normally,  be  represented 
by  Po,  the  factor  i  can  be  calculated  by  means  of  the  expression, 

<-£•  (2) 

where  P  is  the  osmotic  pressure  of  an  abnormal  solute  at  the  same 
concentration.  Since  the  osmotic  pressure  of  a  solution  is  pro- 
portional to  the  relative  lowering  of  its  vapor  pressure,  to  the 
elevation  of  its  boiling-point,  and  to  the  lowering  of  its  freezing- 
point,  it  follows  that  any  one  of  these  colligative  properties  may  be 
similarly  employed  to  calculate  the  value  of  the  empirical  factor, 
i.  A  more  definite  conception  of  the  abnormal  behavior  of  salts 
will  be  gained  by  an  inspection  of  the  following  table  giving  the 
values  of  the  molar  freezing-point  depressions  of  four  typical  salts 
together  with  the  corresponding  values  of  i.  The  headings  of 
the  columns  show  the  number  of  equivalents  of  solute  dis- 
solved iri  1000  grams  of  water. 

As  we  have  learned  in  the  preceding  chapter,  the  formal  freez- 
ing-point depressions  of  dilute  solutions  of  normal  solutes  is  con- 
stant and,  in  the  case  of  aqueous  solutions,  its  value  is  1.86°. 
Inspection  of  the  following  table  reveals  the  fact  that  the  values 
of  the  molar  freezing-point  depressions  of  the  four  salt  solutions 
are  all  greater  than  those  of  solutions  of  normal  solutes,  such  as 
*  Zeit.  phys.  Chem.,  i,  501  (1887). 


212  THEORETICAL  CHEMISTRY 

VALUES  OF  MOLAR  FREEZING-POINT  DEPRESSIONS* 


Concentration 

0.005 

0.006 

0.01 

0.02 

0.05 

3.502 
1.885 
4.776 
2.570 
2.638 
1.420 

0.1 

0.2 

0.5 

KC1                  dTf/N  

3.648 
1.963 
5.308 
2.857 
3.148 
1.694 
6.840 
3.681 

3.640 
1.959 
5.282 
2.843 
3.112 
1.675 
6.810 
3.665 

3.610 
1.943 
5.198 
2.798 
3.006 
1.618 
6.696 
3.604 

3.564 
1.918 
5.040 
2.713 
2.854 
1.536 
6.192 
3.333 

3.451 
1.861 
4.568 
2.459 
2.460 
1.324 

3.394 
1.833 
4.324 
2.333 
2.270 
1.223 

3.314 
1.800 
3.948 
2.316 
2.008 
1.084 

i  

K2SO4               dTf/N 

i.  .  . 

MgSO4              dTf/N  

K3Fe(CN)6       dTf/N.'.'.'.'.'.'.'. 
i  

*  Noyes  and  Falk,  Jour.  Am.  Chem.  Soc.,  32,  1011   (1910). 

cane  sugar  or  mannite.  Furthermore,  it  will  be  seen  that  as  the 
solution^  are  diluted,  their  molar  depressions  approach  limiting 
values  which  are  simple  multiples  of  the  normal  depression,  1.86°. 
Thus,  with  solutions  of  KC1  and  MgS04,  the  molar  depression 
approaches  the  value  3.72°  =  2  X  1.86°,  as  the  solutions  are 
diluted,  while  with  solutions  of  K2SO4  and  K3Fe(CN)6  the  respec- 
tive limits  toward  which  their  formal  depressions  tend  are  5.58°  = 
3  X  1.86°  and  7.44°  =  4  X  1.86°.  The  preceding  examples  being 
typical  of  the  behavior  of  all  solutions  in  which  the  solute  is  an 
acid,  a  base  or  a  salt,  we  may  formulate  the  folllowing  generaliza- 
tion :  —  The  molar  freezing-point  depressions  of  dilute  solutions  of 
all  abnormal  solutes  (acids,  bases  and  salts)  are  greater  than  the  cor- 
responding depressions  of  solutions  of  normal  solutes,  and  in  every 
case,  as  the  solutions  are  diluted,  these  depressians  approach  limit- 
ing values,  which  are  simple  multiples  of  the  normal  depression, 
1.86°. 

These  facts  together  with  numerous  other  properties  of  dilute 
solutions  of  acids,  bases  and  salts  find  their  most  satisfactory 
explanation  in  the  theory  of  electrolytic  dissociation. 

The  Theory  of  Electrolytic  Dissociation.  In  1887,  Arrhenius* 
advanced  an  hypothesis  to  account  for  the  abnormal  osmotic 
activity  of  solutions  of  acids,  bases  and  salts.  He  pointed  out 
that  just  as  the  exceptional  behavior  of  certain  gases  has  been 
completely  reconciled  with  the  law  of  Avogadro,  by  assuming  a 
dissociation  of  the  vaporized  molecule  into  two  or  more  simpler 
molecules,  so  the  enhanced  osmotic  pressure  and  the  abnormally 
*  Zeit.  phys.  Chem.,  i,  631  (1887). 


SOLUTIONS  OF  ELECTROLYTES  213 

great  freezing-point  depressions  of  solutions  of  acids,  bases  and 
salts  can  be  explained,  if  we  assume  a  similar  process  of  dissoci- 
ation. He  proposed,  therefore,  that  aqueous  solutions  of  acids, 
bases  and  salts  be  considered  as  dissociated,  to  a  greater  or  less 
extent,  into  positively-  and  negatively-charged  particles  or  ions, 
and  that  the  increase  in  the  number  of  dissolved  units  due  to  this 
dissociation  is  the  cause  of  the  enhanced  osmotic  activity.  Ac- 
cording to  this  hypothesis,  hydrochloric  acid,  potassium  hydroxide 
and  potassium  chloride,  when  dissolved  in  water,  dissociate  in  the 
following  manner :  — 

HCI  *±  H-  +  cr 

KOH  <=>  K*  +  OH' 

KCI  <=±  K*  +  cr, 

where  the  dots  indicate  positively-charged  ions  and  the  dashes 
negatively-charged  ions.  The  reversed  arrows  indicate  that  a 
definite  equilibrium  exists  between  the  ions  and  the  undissociated 
molecules,  a  certain  number  of  undissociated  molecules  always 
being  present  at  finite  concentrations.  As  the  solution  is  diluted, 
the  percentage  of  undissociated  molecules  steadily  decreases, 
and  ultimately,  when  the  solution  becomes  infinitely  dilute,  the 
solute  is  completely  dissociated. 

In  each  of  the  above  cases,  one  molecule  yields  two  ions,  so 
that,  if  dissociation  is  complete,  the  maximum  osmotic  effect 
should  not  be  greater  than  twice  that  produced  by  an  equimolecu- 
lar  quantity  of  a  substance  which  behaves  normally.  Reference 
to  the  preceding  table  shows  that  the  value  of  i  for  potassium 
chloride  approaches  the  limiting  value  of  2  as  the  solution  is 
diluted.  The  other  salts  given  in  the  table  dissociate,  according 
to  Arrhenius,  in  the  following  manner :  — 

K2SO4<=»2K*  +  SO4", 
MgS04  <=*  Mg"  +  S04", 
K3Fe(CN)6  <=»  3  K*  +  Fe(CN)6'". 

If  these  equations  correctly  represent  the  process  of  dissociation, 
then  when  dissociation  is  complete,  the  osmotic  effect  of  infinitely 
dilute  solutions  of  potassium  sulphate  should  be  three  times  that 
produced  by  an  equimolar  concentration  of  a  normal  solute,  while 
in  the  case  of  an  infinitely  dilute  solution  of  potassium  ferri-cyan- 
ide,  the  effect  should  be  four  times  that  of  a  normal  solute.  On 


214  THEORETICAL  CHEMISTRY 

referring  to  the  table,  it  will  be  seen  that  the  limiting  values  of  i 
for  each  of  the  salts,  KC1,  K2SO4,  MgSO4  and  K3Fe(CN)6,  are  2,  3, 
2  and  4  respectively.  In  other  words,  the  factor,  i,  may  be  re- 
garded as  representing  the  number  of  dissolved  units  resulting 
from  one  formula-weight  of  solute.  It  is  because  of  this  signifi- 
cance which  attaches  to  i  that  is  sometimes  called  the  mol-number 
of  the  solute. 

The  theory  of  electrolytic  dissociation,  or  electrolytic  ioniza- 
tion  as  it  is  frequently  called,  enables  us  to  calculate  the  degree 
of  ionization  in  any  solution  by  comparing  its  freezing-point  de- 
pression with  the  freezing-point  lowering  of  an  equimolecular 
solution  of  a  normal  substance. 

Let  us  suppose  that  the  degree  of  dissociation  of  1  molecule  of 
a  dissolved  substance  is  a,  each  molecule  yielding  n  ions.  Then 
1  —  a  will  be  the  undissociated  portion  of  the  molecule,  and  the 
total  number  of  dissolved  units  will  be 

1  —  a  +  na. 

If  dTf  is  the  depression  of  the  freezing-point  produced  by  the 
substance,  and  dTo/  the  depression  produced  by  an  equimolecular 
quantity  of  an  undissociated  substance,  then 

1  -  a  +  na  =dTL=    . 

1  "  dTv  ~ 


It  will  be  observed  that  this  formula  is  identical  with  that  derived 
for  the  degree  of  dissociation  of  a  gas  (p.  41).  If  this  formula 
be  applied  to  the  freezing-point  data  for  solutions  of  potassium 
chloride  given  in  the  preceding  table,  the  percentage  dissociation 
at  the  different  dilutions  given  in  the  following  table  are  ob- 
tained. 

The  figures  in  the  last  column  show  that  the  degree  of  disso- 
ciation increases  as  the  concentration  diminishes. 

It  was  further  pointed  out  by  Arrhenhis  that  all  of  the  sub- 
stances which  exhibit  abnormal  osmotic  effects,  when  dissolved 
in  water,  yield  solutions  which  conduct  the  electric  current,  where- 
as, aqueous  solutions  of  such  substances  as  cane  sugar,  urea  and 
alcohol,  exert  normal  osmotic  pressures,  but  do  not  conduct  elec- 
tricity any  better  than  the  pure  solvent.  In  other  words,  only 

• 


SOLUTIONS  OF  ELECTROLYTES 


215 


electrolytes*  are  capable  of  undergoing  ionic  dissociation;  hence 
Arrhenius  termed  the  hypothesis  the  electrolytic  dissociation  theory. 

PERCENTAGE  DISSOCIATION  OF  SOLUTIONS  OF 
POTASSIUM  CHLORIDE 


Cone.,  N. 

dTf/N 

t 

a 

0.005 

3.648 

1.963 

0.963 

0.006 

3.640 

1.959 

0.959 

0.01 

3.610 

1.943 

0.943 

0.02 

3.564 

1.918 

0.918 

0.05 

3.502 

1.885 

0.885 

0.1 

3.451 

1.861 

0.861 

0.2 

3.394 

1.833 

0.833 

0.5 

3.314 

1.800 

0.800 

As  we  have  seen,  when  potassium  chloride  is  dissolved  in  water, 
it  is  supposed  to  dissociate  into  positively-charged  potassium  ions 
and  negatively-charged  chlorine  ions.  Accordingly  when  two 
platinum  electrodes,  one  charged  positively  and  the  other 
negatively,  are  introduced  into  the  solution,  the  potassium  ions 
move  toward  the  negative  electrode  and  the  chlorine  ions  move 
toward  the  positive  electrode,  the  passage  of  a  current  through 
the  solution  consisting  in  the  ionic  transfer  of  electric  charges. 
Since  the  undissociated  molecules  are  electrically  neutral  and 
hence  do  not  participate  in  the  transfer  of  electric  charges,  it  fol- 
lows that  the  conductance  of  a  solution  of  an  electrolyte  is  de- 
pendent upon  the  degree  of  dissociation.  The  relation  between 
electrical  conductance  and  the  degree  of  ionization  will  be  dis- 
cussed in  a  subsequent  chapter.  It  may  be  stated  at  this  point, 
however,  that  the  values  of  a,  based  upon  measurements  of  elec- 
trical conductance,  while  showing  some  discrepancies  in  indi- 
vidual cases,  are  in  general  in  good  agreement  with  the  values 
obtained  by  the  freezing-point  method. 

Dissociation  and  Lowering  of  Vapor  Pressure.  The  lowering 
of  the  vapor  pressure  of  water  produced  by  dissolved  potassium 

*  The  term  electrolyte  strictly  refers  to  the  solution  of  an  ionized  substance, 
although  it  is  often  applied  to  acids,  bases  and  salts  because,  when  dissolved, 
they  produce  electrolytes.  To  avoid  confusion,  the  term  "  ionogen  "  (ion 
former)  has  been  proposed  for  those  substances  which  give  conducting  solu- 
tions. 


216 


THEORETICAL  CHEMISTRY 


chloride  has  been  accurately  determined  by  Lovelace,  Frazer 
and  Sease.*  The  following  table  contains  a  summary  of  their 
results,  together  with  the  vapor  pressure  lowerings  for  solutions 
of  mannite  of  equal  concentrations. 

PERCENTAGE  DISSOCIATION  OF  DILUTE   SOLUTIONS  OF  PO- 
TASSIUM CHLORIDE  CALCULATED  FROM   VAPOR 
PRESSURE  LOWERINGS 


Formal 
Concentration 

PI  -  pi 

for  KC1 

PI   -   P2 

for  Mannite 

Percentage  Dissociation 

V.  P.  (20°) 

Cond.  (18°) 

F.-P. 

0.05 
0.10 
0.20 
0.30 
0.40 
0.50 

0.0294 
0.0574 
0.1128 
0.1680 
0.2225 
0.2785 

0.0156 
0.0311 
0.0622 
0.0934 
0.1243 
0.1555 

0.885 
0.846 
0.814 
0.799 
0.790 
0.790 

0.889 
0.860 
0.827 

0.885 
0.861 
0.833 

0^779 

0.800 

By  dividing  the  vapor  pressure  lowerings  of  the  potassium  chlor- 
ide solutions  by  the  corresponding  lowerings  of  the  mannite  solu- 
tions, the  values  of  the  mol-number,  i,  are  obtained,  and  on  sub- 
stituting the  resulting  values  of  i  in  equation  (3),  the  degree  of 
dissociation  given  in  the  fourth  column  of  the  table  can  be  cal- 
culated. The  resulting  values  of  the  degree  of  dissociation  will 
be  seen  to  be  in  fairly  close  agreement  with  those  derived  from 
conductivity  and  freezing-point  measurements. 

Classification  of  Electrolytes.  Electrolytes  may  be  classified 
according  to  the  extent  to  which  they  undergo  ionization  in  solu- 
tion. Those  electrolytes  which  are  largely  dissociated  in  dilute 
solutions  are  commonly  known  as  strong  electrolytes,  whereas 
those  which  are  only  slightly  dissociated  are  called  weak  electro- 
lytes. All  of  the  stronger  acids  and  bases  and  nearly  all  salts 
belong  to  the  class  of  strong  electrolytes. 

Electrolytes  may  also  be  classified  according  to  the  character 
of  the  ions  which  they  yield  in  solution.  Thus,  such  substances 
as  HC1,  KOH,  and  KC1,  which  yield  on  dissociation  two  singly 
charged  ions,  are  called  uni-univalent  electrolytes,  while  such 
substances  as  H2SO4,  Ca(OH)2  and  CaCl2,  which  dissociate  into 
*  Jour.  Am.  Chem.  Soc.,  43,  119  (1921). 


SOLUTIONS  OF  ELECTROLYTES  217 

one  doubly  and  two  singly  charged  ions  are  termed  uni-bivalent 
electrolytes.  Similarly,  electrolytes  such  as  CuSO4  and  MgSO4, 
which  yield  two  doubly  charged  ions,  are  bi-bivalent,  and  K3Fe- 
(CN)6,  which  dissociates  into  one  trebly,  and  three  singly  charged 
ions,  is  uni-trivalent. 

Intermediate  and  Complex  Ions.  Electrolytes  which  yield 
more  than  two  ions  when  completely  dissociated,  probably  under- 
go ionization  in  two  different  ways,  giving  rise  to  so-called  inter- 
mediate ions.  For  example,  in  extremely  dilute  solutions  of 
sulphuric  acid,  dissociation  takes  place  according  to  the  equation, 

H2SO4  <=±  2  IT  +  SO4". 

In  more  concentrated  solutions,  however,  a  portion  of  the  acid 
dissociates  as  above,  and  another  portion  undoubtedly  disso- 
ciates according  to  the  equation, 

H2SO4  <=±  H*  +  HSO/, 

giving  rise  to  the  intermediate  ion,  HSO/.  While  in  the  case  of 
sulphuric  acid,  the  concentration  of  the  intermediate  ion  can  be 
determined,*  we  have  as  yet  no  satisfactory  method  for  estimating 
the  extent  to  which  intermediate  ions  are  formed.  It  is,  there- 
fore, manifestly  impossible  to  determine  the  degree  of  ionization 
in  solutions  of  electrolytes  which  are  capable  of  yielding  inter- 
mediate ions,  except  at  extremely  small  dilutions,  f 

Many  electrolytes  exhibit  a  tendency  to  combine  with  other 
electrolytes  in  solution  to  form  complex  compounds.  When 
these  complex  compounds  dissociate,  they  give  rise  to  complex 
ions.  For  example,  if  a  solution  of  silver  nitrate  is  treated  with 
an  excess  of  potassium  cyanide,  a  reaction  represented  by  the 
following  equation  occurs :  — 

2  KCN  +  AgN03  =  KN03  +  KAg(CN)2. 

The  resulting  complex  electrolyte  dissociates  according  to  the 
equation, 

KAg(CN)2  <=»  K*  +  Ag(CN),', 

giving  rise  to  the  complex  ion,  Ag(CN)2/. 

Properties  of  Completely  Ionized  Solutions.  The  chemical 
properties  of  an  ion  are  very  different  from  the  properties  of  the 

*  See  Noyes  and  Stewart,  Jour.  Am.  Chem.  Soc.,  32,  1133  (1910). 
t  See  Hall  and  Harkins,  ibid.  38,  2672  (1916). 


218  THEORETICAL  CHEMISTRY 

atom  or  radical  when  deprived  of  its  electrical  charge.  For 
example,  the  sodium  ion  is  present  in  an  aqueous  solution  of  so- 
dium chloride,  but  there  is  no  evidence  of  chemical  reaction  with 
the  solvent;  whereas,  the  element  in  the  electrically-neutral  con- 
dition reacts  violently  with  water,  evolving  hydrogen  and  forming 
a  solution  of  sodium  hydroxide.  Again,  take  the  element  chlorine : 
when  chlorine  in  the  molecular  condition,  either  as  gas  or  in  solu- 
tion, is  added  to  a  solution  of  silver  nitrate,  no  precipitate  of 
silver  chloride  is  formed.  Furthermore,  chlorine  in  such  com- 
pounds as  CHCla,  CCU,  etc.,  is  not  precipitated  by  silver  nitrate, 
since  these  compounds  are  not  dissociated  by  water.  Or,  chlorine 
may  be  present  in  a  compound  which  is  dissociated  by  water  and 
yet  not  exhibit  its  characteristic  reactions,  because  it  is  present  in 
a  complex  ion.  Thus,  potassium  chlorate  dissociates  in  the  fol- 
lowing manner :  — 

KC103  <=*  K'  +  CIO  '3. 

On  adding  silver  nitrate,  there  is  no  precipitation,  because  the 
chlorine  forms  a  complex  ion  with  oxygen. 

The  physical  properties  of  completely  ionized  solutions  are,  in 
general,  additive.  This  is  well  illustrated  by  a  series  of  solutions 
of  colored  salts,  the  color  of  which  is  due  to  the  presence  of  a  par- 
ticular ion.  It  is  found,  when  the  solutions  are  sufficiently  dilute 
to  insure  complete  dissociation,  that  they  all  have  the  same  color. 
The  additive  character  of  the  colors  of  solutions  of  electrolytes 
is  brought  out  in  a  striking  manner  by  a  comparison  of  their 
absorption  spectra.  Ostwald*  photographed  the  absorption 
spectra  of  solutions  of  the  permanganates  of  lithium,  cadmium, 
ammonium,  zinc,  potassium,  nickel,  magnesium,  copper,  hydrogen 
and  aluminium,  each  solution  containing  0.002  gram-equivalents 
of  salt  per  liter.  The  absorption  spectra,  as  shown  in  Fig.  70, 
will  be  seen  to  be  practically  identical,  the  bands  occupying  the 
same  position  in  each  spectrum.  This  affords  a  strong  con- 
firmation of  the  theory  of  electrolytic  dissociation,  according  to 
which  a  dilute  solution  is  to  be  regarded  as  a  mixture  of  electrically 
equivalent  quantities  of  oppositely  charged  ions,  each  of  which 
contributes  its  specific  properties  to  the  solution.  The  perman- 
ganate ion  being  colored,  and  common  to  all  of  the  salts  examined, 
and  the  positive  ions  of  the  various  substances  being  colorless, 

*  Zeit.  phys.  Chem.,  9,  579  (1892). 


SOLUTIONS  OF  ELECTROLYTES 


219 


it  follows  that  when  dissociation  is  complete,  the  absorption  spec- 
tra of  all  of  the  solutions  must  be  identical. 

A  number  of  other  properties  of  completely  dissociated  solu- 
tions have  been  shown  to  be  additive.  Among  these  may  be 
mentioned  density,  specific  re- 
fraction, surface  tension,  ther- 
mal expansion,  and  magnetic 
rotatory  power.  Additional 
evidence  in  favor  of  the  theory 
of  electrolytic  dissociation  will 
be  furnished  in  forthcoming 
chapters.  Notwithstanding  the 
large  number  of  facts  which 
can  be  satisfactorily  interpreted 
by  the  theory,  there  are  direc- 
tions in  which  it  requires 
amplification  and  modification. 

Hydration.  Attention  has 
already  been  directed  to  the 
fact,  that  the  lack  of  agree- 
ment between  the  observed  and 
calculated  value  of  the  osmotic 
pressures  of  concentrated  solu- 
tions of  cane  sugar  can  be  sat- 
isfactorily explained  by  assum- 
ing that  each  molecule  of  sugar 
forms  a  pentahydrate  through 
the  removal  of  five  molecules 
of  water  from  the  solvent. 
Additional  evidence  of  the 
existence  of  hydrates  of  cane  sugar  in  aqueous  solution  has  been 
furnished  by  Jones  and  the  author*  from  freezing-point  data,  by 
Scatchardf  from  calculations  based  on  vapor  pressure  measure- 
ments, and  also  by  Philip  J  from  measurements  of  the  solubility 
of  hydrogen  in  sucrose  solutions.  The  average  degree  of  hy- 
dration  of  sugar,  as  calculated  by  Scatchard  from  various  avail- 
able sources,  is  represented  graphically  in  Fig.  71.  The  results 

*  Am.  Chem.  Jour.,  32,  327  (1904). 

f  Jour.  Am.  Chem.  Soc.,  43,  2406  (1921). 

$  Jour.  Chem.  Soc.,  99,  711  (1907). 


Fig.  70 


220 


THEORETICAL  CHEMISTRY 


obtained  by  each  of  the  different  methods  appear  to  agree  within 
the  limits  of  their  respective  experimental  errors  and  to  confirm 
the  existence  of  a  definite  hydrate  of  cane  sugar  in  aqueous  solu- 
tion. 

That  the  phenomenon  of  hydration  is  not  confined  to  non- 
electrolytes  has  been  shown  by  Jones  and  his  students*  who  have 
measured  the  depression  of  the  freezing-point  of  water  produced  by 


°o   o 


Grams  Sucrose  per  100  grams  Water 

Fig.  71 

Full  line:  Hydration  from  vapor  pressure  at  0°. 

Circles:  "  freezing-point,  Jones  and  Getman,  re- 

calculated by  Scatchard. 

Crosses:  "  solubility  of  hydrogen  at  8°,  calculated 

by  Philip. 

Squares:  "  osmotic  pressure  at  0°,  Berkeley,  cal- 

culated by  Porter. 

a  large  number  of  electrolytes  in  very  concentrated  solutions. 
Dissociation  of  electrolytes  in  aqueous  solution  increases  with  the 
dilution,  becoming  complete  at  a  concentration  of  about  0.001 
molar.  We  should  expect  the  dissociation  to  diminish  with  in- 
creasing concentration,  until,  if  the  electrolyte  is  sufficiently 
soluble,  the  depression  of  the  freezing-point  becomes  normal. 
Investigations  by  Jones  and  his  co-workers  f  have  shown  that 
the  facts  are  contradictory  to  this  expectation.  They  found  that 
the  value  of  the  molecular  depression  of  the .  freezing-point  of 
water  produced  by  a  number  of  electrolytes,  diminishes  with  in- 

*  Publ.  No.  60,  Carnegie  Institution. 

t  Am.  Chem.  Jour.  22,  5,  110  (1899);  23,  89  (1900). 


SOLUTIONS  OF  ELECTROLYTES  221 

creasing  concentration  up  to  a  certain  point,  as  would  be 
expected,  and  then  increases  again.  This  is  illustrated  by  the 
freezing-point  curves  for  several  typical  chlorides  shown  in  Fig. 
72.  The  increase  in  the  molecular  depression  becomes  very 
marked  at  great  concentrations;  in  fact,  the  molecular  depression 
in  a  molar  solution  is  frequently  greater  than  the  molar  depression 
corresponding  to  a  completely  dissociated  salt.  This  phenomenon, 
which  has  been  found  to  be  quite  general,  may  be  accounted 
for  by  assuming  that  the  dissolved  substance  has  entered  into 
combination  with  a  portion  of  the  water,  thus  removing  it  from 
the  role  of  solvent.  A  molecular  complex  formed  by  the  union 
of  one  .molecule  of  solute  and  a  number  of  molecules  of  water, 
will  act  as  a  single  dissolved  unit  in  depressing  the  freezing-point 
of  the  pure  solvent.  Evidently  the  total  amount  of  water  pres- 
ent, which  functions  as  solvent,  will  be  diminished  by  the  amount 
of  water  which  has  been  appropriated  by  the  solute. 

If  hydrates  exist  in  aqueous  solution,  then  those  solutes  which 
exhibit  the  greatest  depressing  action  on  the  freezing-point  of 
water,  should  be  the  ones  which  crystallize  from  the  solution . 
with  the  greatest  amount  of  water  of  crystallization.  That 
there  is  a  relation  between  freezing-point  depression  and  water 
of  crystallization,  is  shown  by  the  curves  in  Fig.  72.  Those  salts 
which  crystallize  without  water  of  crystallization  produce  the 
least  depression  of  the  freezing-point.  Those  crystallizing  with 
two  molecules  of  water  are  next  in  order,  and  then  follow  those 
crystallizing  with  four  and  six  molecules  respectively.  Similar 
relations  are  found  to  hold  for  the  corresponding  bromides,  iodides 
and  nitrates.  We  conclude,  therefore,  that  those  solutes  which 
crystallize  with  the  same  amount  of  water  produce  approximately 
the  same  depression  of  the  freezing-point  of  water. 

While  Jones  was  undoubtedly  correct  in  attributing  the  ob- 
served abnormalities  in  the  freezing-point  depressions  of  concen- 
trated solutions  of  electrolytes  to  the  formation  of  hydrates,  his 
method  of  calculating  their  approximate  composition  was  wholly 
erroneous  and,  in  several  cases,  led  to  the  absurd  conclusion  that 
all  of  the  water  present  had  entered  into  combination  with  the 
solute. 

It  is  interesting  to  recall,  that  as  long  ago  as  1893,  Werner* 

*  Zeit.  anorg.  Chem.,  3,  296  (1893). 


222 


THEORETICAL  CHEMISTRY 


expressed  the  view  that  when  an  electrolyte  is  dissolved  in  water, 
hydration  of  the  solute  precedes  its  dissociation,  and,  after  ionic 
equilibrium  has  been  established,  the  positive  ion  will  hold  in 
combination  as  many  molecules  of  water  as  correspond  to  its 


i  2 

Concentration  mols.  per  liter 

Fig.  72 

coordination  number.  For  example,  cupric  chloride  may  be 
considered  to  dissociate  in  dilute  solution  in  the  following 
manner:  — 


CuCl2  +  4  H20 


[H20 
|_H20 


Cu 


H2OT 
H20 


+  2  Cl'. 


The  copper  ion,  it  will  be  observed,  is  assumed  to  be  hydrated  to 
the  extent  of  four  molecules  of  water,  corresponding  to  its  coor- 


SOLUTIONS  OF  ELECTROLYTES  223 

dination  number,  while  the  chlorine  ion  is  assumed  to  remain 
unhydrated. 

The  assumption  that  the  ions  undergo  hydration  in  solution 
has  been  confirmed  experimentally,  but  discussion  of  these  ex- 
periments must  be  deferred  until  a  later  chapter. 

Without  adducing  further  evidence  in  favor  of  the  hydration 
hypothesis,  it  may  be  stated  that  its  validity  is  quite  generally 
accepted,  although  there  are  many  problems  connected  with  the 
hypothesis  which  still  await  solution. 

According  to  the  views  which  have  just  been  outlined,  when 
an  electrolyte  is  dissolved  in  water,  the  molecules  first  become 
hydrated,  and  then  dissociate  into  ions  which  are  hydrated  to 
the  same  or  different  degrees.  For  example,  the  dissociation  of 
potassium  chloride  should  be  expressed  by  the  equation, 

KC1  -f  aq  =  KC1 . aq  =  [K  (H2O)J*  +  [Cl  (H2O)n'] ', 

(solid)  (solution) 

where  the  symbol  aq  denotes  a  large  amount  of  water,  and  where 
n  and  n'  denote  the  numbers  of  molecules  of  water  which  are  asso- 
ciated with  the  two  ions. 

In  the  derivation  of  equation  (3),  expressing  the  relation  be- 
tween freezing-point  depression  and  ionization,  it  was  assumed 
that  the  solute  was  unhydrated.  In  view  of  the  tendency  of  all 
electrolytes  to  become  hydrated  when  dissolved,  it  follows  that 
the  values  of  the  degree  of  ionization,  calculated  from  freezing- 
point  depressions,  must  be  more  or  less  erroneous.  This  question 
has  been  carefully  considered  by  Noyes  and  Falk*  who,  after 
reviewing  the  best  available  experimental  data,  state  that  the 
degree  of  ionization  of  most  uni-univalent  strong  electrolytes, 
in  aqueous  solutions  whose  concentrations  are  less  than  0.1  molar, 
can  be  calculated  from  the  freezing-point  depressions  of  their 
solutions  with  an  accuracy  of  from  1  to  4  per  cent. 

REFERENCES 

The  Electrolytic  Dissociation  Theory  with  Some  of  its  Applications.  Talbot 
and  Blanchard. 

The  Present  Position  of  the  Theory  of  Ionization.  Transactions  of  the  Far- 
aday Society.  15,  1-178,  (1919). 

Osmotic  Pressure.     Findlay,  pp.  58-63. 

Hydrates  in  Solution.     Transactions  of  the  Faraday  Society,     Vol.  Ill  (1907). 

*  loc.  cit. 


224  THEORETICAL  CHEMISTRY 

.  PROBLEMS 

1.  At  18°  C.  a  0.5  molar  solution  of  NaCl  is  74.3  per  cent  dissociated. 
What  would  be  the  osmotic  pressure  of  the  solution  in  atmospheres  at 
18°  C.?  Ans.  20.79. 

.^2.  The  vapor  pressure  of  water  at  20°  C.  is  17.406  mm.  and  that  of  a 
0.2  molar  solution  of  potassium  chloride  is  17.296  mm.  at  the  same  tem- 
perature. Calculate  the  degree  of  dissociation  of  the  salt. 

Ans.  75.38  per  cent. 

^3.  The  degree  of  dissociation  of  a  0.5  molar  solution  of  sodium  chloride 
at  25°  is  74.3  per  cent.  Calculate  the  osmotic  pressure  of  the  solution  at 
the  same  temperature.  Ans.  21.32  atmos. 

^/4.  A  solution  containing  1.9  mols  of  calcium  chloride  per  liter  exerts 
the  same  osmotic  pressure  as  a  solution  of  glucose  containing  4.05 
mols  per  liter.  What  is  the  degree  of  ionization  of  the  calcium  chloride? 

Ans.  56.6  per  cent. 

5.  At  0°  C.  the  vapor  pressure  of  water  is  4.620  mm.  and  of  a  solution 
of  8.49  grams  of  NaNO3  in  100  grams  of  water  4.483  mm.     Calculate  the 
degree  of  ionization  of  NaN03.  Ans.  64.9  per  cent. 

6.  At  0°  C.  the  vapor  pressure  of  water  is  4.620  mm.  and  that  of  a 
solution  of  2.21  grams  of  CaCl2  in  100  grams  of  water  is  4.583  mm. 
Calculate  the  apparent  molecular  weight  and  the  degree  of  ionization  of 
CaCl2.  Ans.  M  =  49.66,  «  =  62  per  cent. 

7.  The  boiling-point  of  a  solution  of  0.4388  gram  of  sodium  chloride 
in  100  grams  of  water  is  100°. 074  C.     Calculate  the  apparent  molecular 
weight  of  the  sodium  chloride  and  its  degree  of  ionization.     Kb   = 
0.518.  Ans.  M  =  30.84,  a  =  89.7  per  cent. 

8.  The  boiling-point  of  a  solution  of  3.40  grams  of  BaCl2  in  100  grams 
of  water  is  100°.208  C.    Kb  =  0.518.    What  is  the  degree  of  ionization 
of  the  BaCl2?  Ans.  72.5  per  cent. 

9.  At  100°  C.  the  vapor  pressure  of  a  solution  of  6.48  grams  of  ammo- 
nium chloride  in  100  grams  of  water  is  731.4  mm.     Kb  =  0.518.    What 
is  the  boiling-point  of  the  solution?  Ans.  101.09°  C. 

>/ 10.  A  solution  of  1  gram  of  silver  nitrate  in  50  grams  of  water  freezes 
at  — 0°.348  C.  Calculate  to  what  extent  the  salt  is  ionized  in  solution. 
Kf  =  1.86.  Ans.  59  per  cent. 

4  11.  A  solution  of  NaCl  containing  3.668  grams  per  1000  grams  of 
water  freezes  at  — 0°.2207  C.  Calculate  the  degree  of  ionization  of  the 
salt.  Kf  =  1.86.  Ans.  89.2  per  cent. 

12.  The  freezing-point  of  a  solution  of  barium  hydroxide  containing 
1  mol  in  64  liters  is  -0°.0833  C.     What  is  the  concentration  of  hydroxyl 
ions  in  the  solution?     Take  Kf  —  1.86  for  concentrations  in  mols  per 
liter.  Ans.  0.0284  gm.  -  ion  per  liter. 

13.  The  vapor  pressure  of  water  at  0°  C.  is  4.620  mm.,  and  the  lower- 


SOLUTIONS  OF  ELECTROLYTES  225 

ing  of  the  vapor  pressure  produced  by  dissolving  5.64  grams  of  sodium 
chloride  in  100  grams  of  water  is  0.142  mm.  What  is  the  freezing-point 
of  the  solution?  Kf  =  1.86.  Ans.  -  3.177°  C. 

14.  A  solution  containing  8.34  grams  Na2S04  per  1000  grams  of  water 
freezes  at  — 0°.280  C.  Assuming  dissociation  into  3  ions,  calculate  the 
degree  of  ionization  and  the  concentrations  of  the  Na*  and  SO4"  ions. 
Kf  =  1.86.  Ans.  a  =  78.2  per  cent;  Na  =  0.0918  gm.  -  ion  per  liter; 
S04"  =  0.0459  gm.  -  ion  per  liter. 


CHAPTER  X 
COLLOIDS 

Crystalloids  and  Colloids.  In  the  course  of  his  investigations 
on  diffusion  in  solutions,  Thomas  Graham*  drew  a  distinction 
between  two  classes  of  solutes,  which  he  termed  crystalloids  and 
colloids.  Crystalloids,  as  the  name  implies,  can  be  obtained  in  the 
crystalline  form :  to  this  class  belong  nearly  all  of  the  acids,  bases 
and  salts.  Colloids,  on  the  other  hand,  are  generally  amorphous, 
such  substances  as  albumin,  starch  and  caramel  being  typical  of 
the  class.  Because  of  the  gelatinous  character  of  many  of  the 
substances  belonging  to  this  class,  Graham  termed  them  colloids 
(/coXXa  =  glue,  and  eidos  =  form).  The  differences  between  the 
two  classes  are  most  apparent  in  the  physical  properties  of  their 
solutions.  Crystalloids  diffuse  much  more  rapidly  than  colloids; 
thus,  the  velocity  of  diffusion  of  caramel  is  nearly  100  times 
slower  than  that  of  hydrochloric  acid  at  the  same  tempera- 
ture. While  crystalloids  exert  osmotic  pressure,  lower  the  vapor 
pressure  and  depress  the  freezing-point  of  the  solvent,  colloids 
have  very  little  effect  upon  the  properties  of  the  solvent.  The 
marked  difference  between  the  rate  of  diffusion  of  crystalloids,  on 
the  one  hand,  and  that  of  colloids,  on  the  other,  renders  their 
separation  comparatively  easy. 

If  a  solution  containing  both  crystalloids  and  colloids  be 
placed  in  a  vessel  over  the  bottom  of  which  is  stretched  a  col- 
loidal membrane,  such  as  parchment,  and  the  whole  be  im- 
mersed in  pure  water,  the  crystalloids  will  pass  through  the 
membrane,  while  the  colloids  will  be  left  behind.  This  process 
was  termed  by  Graham,  dialysis,  while  the  apparatus  employed 
to  effect  such  a  separation  was  called  a  dialyzer.  When  a  solution 
of  sodium  silicate  is  added  to  an  excess  of  hydrochloric  acid,  the 
products  of  the  reaction,  silicic  acid  and  sodium  chloride,  remain 
in  solution.  When  the  mixture  is  placed  in  a  dialyzer,  the 
sodium  chloride  and  the  hydrochloric  acid,  being  crystalloids, 

*  Lieb.  Ann.,  xai,  1  (1862). 
226 


COLLOIDS  227 

diffuse  through  the  membrane  of  the  dialyzer,  leaving  behind  the 
colloidal  silicic  acid. 

The  terms  crystalloid  and  colloid,  as  used  at  the  present  time, 
have  acquired  different  meanings  from  those  assigned  to  them  by 
Graham.  The  terms  are  now  considered  to  refer,  not  so  much  to 
different  classes  of  substances,  as  to  different  states  which  almost 
all  substances  can  assume  under  certain  conditions. 

Disperse  Systems.  If  in  a  two-component  system,  the  size  of 
the  particles  of  one  of  the  components  be  gradually  reduced  until 
its  dimensions  become  microscopic  or  submicroscopic,  we  have 
what  is  known  as  a  disperse  system.  The  characteristic  properties 
of  disperse  systems  are  attributable  to  the  enormous  surface  of 
the  dispersed  phase.  So-called  "colloidal  solutions"  are  in  reality 
examples  of  such  disperse  systems. 

Nomenclature.  Graham  distinguished  between  two  condi- 
tions in  which  colloids  were  obtainable,  the  term  sol  being  applied 
to  forms  in  which  the  system  resembled  a  liquid,  while  the  term 
gel  was  used  to  designate  those  forms  which  were  solid  and  jelly- 
like.  When  one  of  the  components  of  the  solution  was  water,  the 
two  forms  were  called  a  hydrosol  and  a  hydrogel.  In  like  manner, 
when  alcohol  was  one  of  the  components,  the  terms  alcosol  and 
alcogel  were  applied  to  the  two  forms. 

As  the  knowledge  of  colloids  has  developed,  it  has  become  neces- 
sary to  supplement  Graham's  nomenclature  by  the  introduction 
of  various  other  terms.  It  is  known  to-day  that  the  essential 
difference  between  colloidal  suspensions  and  solutions  on  the  one 
hand,  and  true  solutions  on  the  other,  is  due  to  the  difference  in  the 
degree  of  subdivision,  or  degree  of  dispersion  of  the  dissolved  sub- 
stance. In  a  true  solution,  the  dissolved  substance  is  generally 
present  either  in  the  molecular  or  ionic  condition,  as  may  be  shown 
by  means  of  the  familiar  osmotic  methods  for  molecular  weight 
determination.  In  colloidal  solutions,  however,  the  degree  of 
dispersion  is  not  so  great,  it  having  been  found  to  vary  from  above 
the  limit  of  microscopic  visibility,  (1  X  10~5  cm.),  to  that  of  molec- 
ular dimensions,  (1  X  10~8  cm.).  When  the  degree  of  dispersion 
varies  from  1  X  10~3  cm.  to  1  X  10~5  cm.,  the  particles  are  termed 
microns.  The  properties  of  the  disperse  phase  at  this  degree  of 
dispersion  differ  appreciably  from  the  properties  of  the  same  sub- 
stance when  present  in  large  masses.  When  the  degree  of  dis- 
persion lies  between  1  X  10~5  cm.  and  5  X  10~7  cm.  the  particles 


228  THEORETICAL  CHEMISTRY 

are  known  as  submicrons.  The  existence  of  particles  whose  diam- 
eters are  approximately  1  X  10~7  cm.  has  been  demonstrated  by 
Zsigmondy  with  the  ultramicroscope ;  these  minute  particles  are 
termed  amicrons.  When  the  degree  of  dispersion  is  increased 
beyond  this  limit,  all  heterogeneity  apparently  vanishes,  and  we 
enter  the  realm  of  true  solutions. 

When  the  dispersion  is  not  too  great,  colloidal  solutions  may  be 
divided  into  suspensions  and  emulsions,  according  to  whether  the 
disperse  phase  is  a  solid  or  a  liquid.  As  the  dispersion  is  increased, 
we  obtain  suspension  and  emulsion  colloids,  which  may  be  con- 
veniently called  suspensoids  and  emulsoids.  Suspensoids  and 
emulsoids  are  included  under  the  general  term  dispersoids.  In 
certain  cases,  although  the  disperse  phase  is  unquestionably  liquid, 
the  systems  resemble  suspensoids  in  their  behavior,  while  in  other 
cases,  where  the  disperse  phase  is  solid,  the  systems  exhibit  proper- 
ties characteristic  of  emulsoids.  For  this  reason,  the  classification 
of  sols  as  suspensoids  and  emulsoids  is  not  entirely  satisfactory. 
A  better  system  is  that  in  which  the  presence  or  the  absence  of 
affinity  between  the  disperse  phase  and  the  dispersion  medium  is 
made  the  basis  of  classification.  Where  there  is  marked  affinity 
between  the  two  phases,  the  system  is  termed  lyophile,  and  where 
such  affinity  is  absent,  the  system  is  termed  lyophobe.  When  the 
dispersion  medium  is  water,  the  terms  hydrophile  and  hydrophobe 
are  employed. 

In  the  reversible  transformation  of  a  sol  into  a  gel,  we  are 
not  warranted  in  referring  to  the  change  from  gel  to  sol  as 
an  act  of  solution,  for  if  the  gel  really  dissolved,  a  solution 
and  not  a  sol  would  result.  Various  terms  have  been  proposed 
for  these  reversible  transformations  but  perhaps  the  most 
satisfactory  are  the  terms  gelation  and  solation,  the  former 
designating  the  formation  of  a  gel  from  a  sol  and  the  latter 
the  reverse  process. 

The  Ultramicroscope.  When  a  narrow  beam  of  sunlight  is 
admitted  into  a  darkened  room,  the  dust  particles  in  its  path  are 
rendered  visible  by  the  scattering  of  the  light  at  the  surface  of  the 
particles.  If  the  air  of  the  room  is  free  from  dust,  no  shining 
particles  will  be  seen  and  the  space  is  said  to  be  "  optically  void." 
When  the  particles  of  dust  are  very  minute,  the  beam  of  light 
acquires  a  bluish  tint. 

The  luminosity  of  the  path  of  a  beam  of  light  is  known  as  the 


COLLOIDS 


229 


Tyndall  phenomenon  and  may  be  considered  as  an  indication 
of  the  presence  of  suspended  particles,  .  provided  the  luminosity 
is  not  caused  by  fluorescence.  Almost  all  colloidal  solutions 
exhibit  this  phenomenon  when  a  powerful  beam  of  light  is 
passed  through  them,  thus  proving  the  presence  of  discrete 
particles  in  the  solutions. 

The  ultramicroscope  is  an  instrument  devised  by  Siedentopf 
and  Zsigmondy*  for  the  detection  of  colloidal  particles  much  too 
small  to  be  seen  by  the  naked  eye.  A  powerful  beam  of  light, 
issuing  from  a  horizontal  slit,  is  brought  to  a  focus  within  the 
colloidal  solution  under  examination  by  means  of  a  microscope 
objective,  and  this  image  is  viewed  through  a  second  microscope, 
the  axis  of  which  is  at  right  angles  to  the  path  of  the  beam. 

When  examined  in  this  way,  a  colloidal  solution  appears  to  be 
swarming  with  brilliantly  colored  particles,  moving  rapidly  in  a 
dark  field;  whereas  a  true 
solution,  if  properly  prepared, 
appears  optically  void.  With 
the  ultramicroscope  it  is  pos- 
sible to  count  the  number  of 
particles  present  in  a  given 
volume  of  a  colloidal  solution. 
By  means  of  a  chemical  ana- 
lysis, the  mass  of  colloid  per 
unit  of  volume  can  be  deter- 
mined, and  from  this,  the 
average  mass  of  each  particle 
can  be  calculated.  If  the 
particles  be  assumed  to  be 
spherical  in  shape  and  to 
have  the  same  density  as 
larger  masses  of  the  same 
substance,  we  can  calculate 
the  volume  of  a  single  particle,  and  from  this  its  diameter.  Thus, 
Burtonf  in  his  experiments  on  gold,  silver  and  platinum  sols, 
found  the  average  diameter  of  the  colloidal  particles  to  range 
from  0.2  to  0.6  micron. 


73 


*  "  Colloids  and  the  Ultramicroscope,"   by  R.   Zsigmondy. 
Alexander.     John  Wiley  &  Sons,  Inc. 
t  Phil.  Mag.,  n,  425  (1906). 


Trans,  by 


230  THEORETICAL  CHEMISTRY 

Zsigmondy's  latest  ultramicroscope,  Fig.  73,  consists  of  two 
compound  microscopes  placed  at  right  angles,  and  having  their 
objectives  so  cut  away  as  to  permit  them  to  be  brought  very  close 
together.  A  drop  of  the  liquid  to  be  examined  is  placed  between 
the  front  lenses  of  the  objectives  where  it  is  held  by  capillary 
attraction,  thus  doing  away  with  the  necessity  for  a  cell  as  in  the 
earlier  form  of  instrument.  Since  the  rays  of  light  have  only  a 
short  distance  to  travel,  it  is  possible  to  examine  dark  as  well  as 
colorless  liquids.  With  this  most  improved  type  of  ultramicro- 
scope, it  is  possible  for  the  observer  to  discern  particles  whose 
diameters  range  from  1  to  2  milli-microns. 

The  ultramicroscopic  character  of  emulsoids  is  by  no  means 
sharply  defined,  notwithstanding  the  fact  that  they  exhibit  the 
Tyndall  phenomenon.  It  has  been  suggested  by  Zsigmondy,  that 
the  lack  of  sharpness  in  definition  observed  with  emulsoids  is 
probably  due  to  the  relatively  small  difference  between  the  re- 
fractive indices  of  the  disperse  phase  and  the  dispersion  medium. 
Where  the  difference  between  the  refractive  indices  of  the  two 
phases  is  very  great,  as  in  the  case  of  the  metallic  sols,  excellent 
definition  is  obtained.  It  is  of  interest  to  note,  that  although  the 
basic  hydroxides  are  apparently  suspensoids,  yet  in  their  ultra- 
microscopic  characteristics  they  closely  resemble  emulsoids. 

Ultrafiltration.  Almost  all  sols  can  be  filtered  through  ordinary 
filter  paper  without  undergoing  more  than  a  slight  change  in  con- 
centration, due  to  initial  adsorption.  The  rate  of  filtration  varies 
widely,  depending  upon  the  viscosity  of  the  sol.  As  a  general 
rule,  emulsoids  filter  more  slowly  than  suspensoids,  owing  to  the 
high  viscosity  of  the  former. 

By  filtering  an  arsenious  sulphide  sol  through  a  porous  earthen- 
ware filter,  Linder  and  Picton*  succeeded  in  obtaining  four  differ- 
ent sizes  of  particles  which  they  described  as  follows:  —  (1)  visible 
under  the  microscope,  (2)  exhibited  the  Tyndall  phenomenon,  (3) 
retained  by  porous  plate,  and  (4)  passed  through  porous  plate  un- 
changed. By  employing  plates  of  different  degrees  of  porosity, 
and  determining  the  average  size  of  the  pores  which  just  permit 
filtration,  it  is  possible  to  determine  the  size  of  the  particles  which 
constitute  the  disperse  phase  of  a  sol.  If  we  make  use  of  a  series 
of  graduated  filters,  prepared  by  impregnating  filter  paper  with  a 

*  Jour.  Chem.  Soc.,  61,  148  (1892). 


COLLOIDS  231 

solution  of  collodion  in  acetic  acid,  it  is  not  only  possible  to  sepa- 
rate suspensoids  from  their  dispersion  media,  but  also  to  effect 
the  concentration  of  emulsoids.  Furthermore,  such  filters  are 
useful  in  removing  impurities  from  sols,  the  impurity  passing 
through  the  filter  in  a  manner  similar  to  the  passage  of  the  solvent 
through  the  membrane  in  the  process  of  dialysis.  Ultrafiltration 
is  an  exceedingly  complex  process,  involving  the  phenomena  of 
adsorption  and  dialysis,  in  addition  to  the  ordinary  process  of 
mechanical  separation.  The  complexity  of  the  process  is  well 
illustrated  by  the  phenomena  attendant  upon  the  filtration  of 
almost  any  positive  hydrosol.  Thus,  if  we  attempt  to  filter  a 
ferric  hydroxide  hydrosol  through  a  porous  plate,  or  even  through 
an  ordinary  filter  paper,  we  shall  find  that  the  colloid  will  be  par- 
tially retained  by  the  filter.  This  is  due  to  the  fact  that  the  filter 
becomes  negatively  charged  in  contact  with  water  and,  on  the 
entrance  of  the  positively-charged  sol  into  the  pores  of  the  filter, 
the  colloid  is  immediately  discharged  and  the  disperse  phase  pre- 
cipitated. After  the  pores  of  the  filter  become  partially  stopped 
with  particles  of  the  colloid,  the  sol  will  then  pass  through  un- 
changed. 

Classification  of  Dispersoids.  There  is  abundant  evidence  in 
favor  of  the  view,  that  colloidal  solutions  and  simple  suspensions 
are  closely  related.  Suspensions  of  all  grades  exist,  from  those  in 
which  the  suspended  particles  are  coarse-grained  and  visible  to 
the  naked  eye,  down  to  those  in  which  a  high-power  microscope 
is  required  to  render  the  suspended  particles  visible.  Colloidal 
solutions  have  also  been  shown  to  be  non-homogeneous,  the 
presence  of  discrete  particles  being  revealed  by  means  of  the 
ultramicroscope.  It  follows,  therefore,  that  the  size  of  the  parti- 
cles in  solution  determines  whether  a  substance  is  to  be  considered 
as  a  colloid  or  not.  At  one  extreme,  we  have  true  solutions  in 
which  no  lack  of  homogeneity  can  be  detected,  even  by  the  ultra- 
microscope,  and  at  the  other  extreme,  we  have  coarse-grained 
suspensions,  in  which  the  particles  are  visible  to  the  naked  eye. 
Between  these  two  limits  all  possible  degrees  of  subdivision  are 
possible,  and  it  is  a  very  difficult  matter  to  draw  sharp  lines  of  dis- 
tinction between  true  solutions  and  colloidal  solutions,  on  the  one 
hand,  and  between  colloidal  solutions  and  suspensions,  on  the 
other. 

One  of  the  most  satisfactory  schemes  of  classification  is  that 


232 


THEORETICAL  CHEMISTRY 


of  von  Weimarn  and  Wo.  Ostwald.*  Because  of  the  fact  that 
suspensions,  colloidal  solutions,  and  true  solutions  represent  vary- 
ing degrees  of  dispersion  of  the  solute,  all  three  types  of  systems 
are  termed  by  these  authors,  dispersoids.  The  dispersoids  are 
classified  as  shown  in  the  accompanying  diagram. 

DISPERSOIDS 


Coarse  disp 

ersions  (sus- 

pensions, 

emulsions, 

etc.). 

Magnitude 

of  particles 

greater  than  0.1  »* 

Colloidal  solutions.        Molecular             Ionic 

Magnitude  of  particles     dispersoids.     dispersoids. 

between  0.1  p.  and  1  ju/z- 

Decreasing  degree  of      Magnitude   of   particles, 
"colloidity."  about  1  MM  and  smaller. 

Increasing  degree  of  dis- 
persion. 

•10  =  1  micron  =  0.001  mm. 

The  Brownian  Movement.  If  a  liquid  in  which  fine  particles 
of  matter  are  suspended,  such  as  an  aqueous  suspension  of  gam- 
boge, be  examined  under  the  microscope,  the  suspended  particles 
will  be  seen  to  be  in  a  state  of  ceaseless,  erratic  motion.  This 
phenomenon,  which  was  first  observed  in  1827  by  the  English 
botanist,  Robert  Brown,  while  examining  a  suspension  of  pollen 
grains,  is  known  as  the  Brownian  Movement. 

Ever  since  its  discovery,  the  Brownian  Movement  has  been  the 
subject  of  numerous  investigations.  It  was  not  until  1863,  how- 
ever, that  Wiener  suggested  that  the  cause  of  the  phenomenon  was 
the  actual  bombardment  of  the  suspended  particles  by  the  mole- 
cules of  the  suspending  medium.  Twenty-five  years  later,  a  sim- 
ilar conclusion  was  reached  independently  by  Gouy,  who  showed 
that  neither  light  nor  convection  currents  within  the  liquid  could 
possibly  give  rise  to  the  motion.  Furthermore,  Gouy  showed  the 
movement  to  be  independent  of  external  vibration,  and  only 
slightly  influenced  by  the  nature  of  the  suspended  particles.  The 

*  Roll.  Zeitschrift,  3,  26  (1908), 


COLLOIDS  233 

smaller  the  particles  and  the  less  viscous  the  suspending  medium, 
the  more  rapid  the  motion  was  found  to  be.  By  far  the  most 
striking  feature  of  the  phenomenon,  however,  is  the  fact  that  the 
motion  is  ceaseless. 

Perrin'  s  Experiments.  The  first  quantitative  investigation 
of  the  Brownian  Movement  was  undertaken  by  Perrin  in  1909. 
It  has  been  shown  (p.  49),  that  the  mean  kinetic  energy  Et,  of  one 
mol  of  a  perfect  gas  is  given  by  the  expression, 

Et  =  %pv.  (1) 

Since  pv  =  RT,  we  may  write 

' 


where  N  denotes  the  Avogadro  Constant,  that  is,  the  number  of 
molecules  contained  in  one  mol  of  any  gas.  It  is  evident  that  if 
Et  can  be  measured,  equation  (2)  affords  a  means  of  calculating 
Nt  provided  we  are  warranted  in  applying  an  equation  which  has 
been  derived  for  the  gaseous  state,  to  a  suspension  of  fine  particles 
in  a  liquid  medium.  It  has  already  been  shown,  that  the  simple  gas 
laws  hold  for  dilute  solutions,  and  therefore  we  may  assume  that, 
at  the  same  temperature,  the  mean  kinetic  energy  of  the  dissolved 
molecules  is  equal  to  that  of  the  gaseous  molecules.  In  other 
words,  at  the  same  temperature,  the  mean  kinetic  energy  of  the 
molecules  of  all  fluids  is  the  same,  and  is  directly  proportional 
to  the  absolute  temperature.  Since  the  gas  laws  apply  equally  well 
to  dilute  solutions,  containing  either  large  or  small  molecules, 
Perrin  held  that  there  was  no  d  priori  reason  for  assuming  that  the 
grains  of  a  suspension  should  not  conform  to  the  same  laws.  If 
this  assumption  be  correct,  the  grains  of  a  uniform  suspension 
should  so  distribute  themselves  under  the  influence  of  gravity 
that,  when  equilibrium  is  attained,  the  lower  layers  will  have  a 
higher  concentration  than  the  upper  layers.  In  other  words,  the 
distribution  should  be  strictly  analogous  to  the  distribution  of  the 
air  over  the  surface  of  the  earth,  the  density  being  greatest  at  the 
surface  and  diminishing  as  the  altitude  increases. 

Let  us  imagine  a  suspension  to  be  confined  within  a  tall 
vertical  cylinder  whose  cross-sectional  area  is  s  sq.  cm.  Assum- 
ing that  the  suspension  has  come  to  equilibrium  under  the  in^~ 


234  THEORETICAL  CHEMISTRY 

fluence  of  gravitation,  let  n  be  the  number  of  grains  per  unit  of 
volume,  at  a  height  h  from  the  base  of  the  cylinder.  Since  the 
concentration  diminishes  as  the  height  increases,  the  number  of 
grains  at  a  height  h  +  dh  will  be  n  —  dn.  The  osmotic  pressure 
of  the  grains  at  the  height  h,  will  be  f  nE*,  where  Et  is  the  mean 
kinetic  energy  of  each  grain.  In  like  manner,  the  osmotic  pres- 
sure at  the  height  h  +  dh  will  be  f  (n  —  dn)Et.  The  difference  in 
osmotic  pressure  between  the  two  levels  is  —  f  dnEt  and,  since  the 
pressure  acts  over  a  surface  of  s  sq.  cm.,  the  difference  of  osmotic 
forces  acting  over  the  cross-sectional  area  of  the  cylinder  is 
—  f  s  duEk,  Since  the  system  is  in  equilibrium,  this  difference  in 
osmotic  forces  must  be  balanced  by  the  difference  in  the  attrac- 
tion of  gravitation  at  the  two  levels.  Let  <£  be  the  volume  of  a 
single  grain,  D  its  density,  and  5  the  density  of  the  suspending 
medium.  The  resultant  downward  pull  upon  a  single  grain  will 
be  4>  (D  —  5)  g,  where  g  is  the  acceleration  due  to  gravity.  The 
volume  of  liquid  between  the  two  levels  being  sdh,  it  follows  that 
the  total  downward  pull  upon  all  the  grains  included  between  the 
two  levels  must  be,  nsdh<f>  (D  —  5)  g.  It  is  this  force  which  opposes 
the  tendency  of  the  grains  to  distribute  themselves  uniformly 
throughout  the  entire  volume  of  the  suspending  medium,  or,  in 
other  words,  it  is  the  force  which  acts  in  opposition  to  the  osmotic 
force  —  f  s  dnEt.  When  equilibrium  is  established,  these  two 
forces  must  be  equal,  and  we  may  then  write 

-  |  s  dnEt  =  nsdh<f>  (D  -  6)  g.  (3) 

If  n0  and  n,  denote  the  number  of  grains  per  unit  of  volume  at  each 

of  two  planes,  h  units  apart,  we  obtain,  on  integrating  equation  (3), 

f  Ek  loge  nQ/n  =  0  (D  -  5)  gh.  (4) 

On  substituting  in  equation  (4),  the  value  of  Et  in  equation  (2), 
and  transforming  to  Briggsian  logarithms,  we  have 

2.303  RT/N  log  n0/n  =  f  irr*g  (D  -  d)  h,  (5) 

0  being  expressed  in  terms  of  the  mean  radius,  r,  of  a  single 
grain.  It  is  evident  that  if  we  can  measure  n,  nQ)  D,  and  r  in 
equation  (5),  the  calculation  of  the  Avogadro  Constant,  N,  be- 
comes possible.* 

*  It  has  been  pointed  out  by  Burton,  that  Perrin,  in  the  derivation  of  his 
equation  for  the  distribution  of  the  suspended  particles  under  gravity,  failed 
to  take  into  consideration  the  fact  that  they  are  electrically  charged. 


COLLOIDS  235 

The  determination  of  the  density  of  the  grains,  D,  was  carried 
out  in  two  different  ways  with  suspensions  of  gamboge  and  mastic 
which  had  been  rendered  uniform  by  a  process  of  centrifuging. 
In  the  first  method,  the  grains  were  dried  to  constant  weight 
at  110°,  and  then  by  heating  to  a  higher  temperature,  a  viscous 
liquid  was  obtained  which,  on  cooling,  formed  a  glassy  solid.  The 
density  of  this  solid  was  determined  by  suspending  it  in  a  solution 
of  potassium  bromide  of  known  density. 

In  the  second  method  for  the  determination  of  D,  Perrin  meas- 
ured the  masses,  mi  and  m2,  of  equal  volumes  of  water  and  sus- 
pension, respectively.  On  evaporating  the  suspension  to  dryness, 
the  mass  w3,  of  suspended  solid  contained  in  w2  grams  of  sus- 
pension was  obtained.  If  the  density  of  water  is  dt  the  volume  of 
the  suspended  grains  will  be, 

T7      mi      mz  —  ra3  ,„. 

'-~-    ~-'  6 


and  consequently,  the  density  of  the  grains  will  be,  m^/V.  The 
values  of  D,  obtained  by  these  two  methods,  were  found  to  be  in 
excellent  agreement. 

A  microscope,  furnished  with  suitable  micrometers,  was  employed 
in  the  determination  of  n  and  n0.  With  the  high  magnification 
employed,  the  depth  of  the  field  of  view  was  limited:  in  fact,  the 
measurements  were  carried  out  with  a  microscope-slide  similar 
to  those  used  for  counting  the  corpuscles  in  the  blood.  By  focus- 
ing the  microscope  at  different  depths,  the  average  number  of 
grains  in  the  field  of  view,  at  each  level,  could  be  counted.  Perrin 
was  able  to  photograph  the  larger  grains  at  different  levels,  where- 
as, with  the  smaller  grains,  it  was  necessary  to  reduce  the  field  so 
that  relatively  few  grains  were  visible.  The  average  number  of 
grains  counted  at  any  two  different  levels  would  of  course  give  the 
desired  ratio,  UQ/U. 

The  only  other  quantity  in  equation  (5)  to  be  measured  is  the 
average  radius  of  the  grains,  r.  To  determine  this  quantity, 
Perrin  made  use  of  a  method  similar  to  that  used  by  Thomson  for 
counting  the  number  of  electrically  charged  particles  in  an  ionized 
gas.  Stokes  has  shown  that  the  force  required  to  impart  a  uni- 
form velocity  v,  to  a  particle  of  radius  r,  moving  through  a  liquid 
medium  whose  viscosity  is  rj,  is  given  by  the  formula,  6  irrjrv.  If 


236  THEORETICAL  CHEMISTRY 

the  motion  be  due  to  gravity,  as  in  the  case  of  suspensions  of  fine 
particles,  obviously  the  foregoing  expression  must  be  equal  to  the 
right-hand  side  of  equation  (5),  or 

6  irvfrv  =  I  Trr3  (D  -  6)  g.  (7) 

From  this  equation,  the  value  of  r  can  be  calculated.  The  rate 
at  which  the  grains  settled  under  the  influence  of  gravity  was 
determined  by  placing  a  portion  of  the  uniform  suspension  in  a 
capillary  tube,  and  observing  the  rate  at  which  the  suspension 
cleared,  care  being  taken  to  keep  the  temperature  constant.  This 
method  of  determining  r,  is  open  to  the  objection  that  Stokes' 
law  might  not  apply  to  particles  as  small  as  those  of  colloidal 
suspensions. 

In  order  to  test  the  validity  of  Stokes'  law  under  these  condi- 
tions, the  following  modification  of  the  method  for  the  determina- 
tion of  r,  was  introduced.  It  had  been  observed  that  when  a  sus- 
pension is  rendered  slightly  acid,  the  grains,  on  coming  in  con- 
tact with  the  walls  of  the  containing  vessel,  adhered,  while  the 
motion  of  the  grains  throughout  the  bulk  of  the  liquid  remained 
unaltered.  In  this  way  it  was  possible  to  gradually  remove  all  of 
the  grains  from  the  suspension  and  count  them,  and  knowing  the 
total  volume  of  suspension  taken,  the  average  number  of  grains 
per  cubic  centimeter  could  be  calculated.  If  the  total  mass  of 
suspended  matter  is  known  it  is  an  easy  matter  to  calculate  the 
volume  of  each  grain,  and  from  this  to  compute  the  radius,  r.  The 
value  of  r,  determined  in  this  way,  was  found  to  agree  with  that 
calculated  by  the  first  method,  thus  proving  the  validity  of  Stokes' 
law  when  applied  to  colloidal  suspensions. 

Five  series  of  experiments  carried  out  by  Perrin  with  gamboge 
suspensions  in  which  several  thousand  individual  grains  were 
counted,  gave  as  a  mean  value  of  N,  in  equation  (5),  69  X  1022. 
Similar  experiments  with  mastic  suspensions  gave  N  =  70.0  X 
1022.  These  values,  it  will  be  found  to  be,  in  close  agreement  with 
the  values  of  Avogadro's  Constant  given  on  page  24. 

The  Law  of  Molecular  Displacement.  The  actual  movements 
of  the  individual  grains  of  a  suspension  when  observed  under  the 
microscope  are  seen  to  be  exceedingly  complex  and  erratic.  The 
horizontal  projections  of  the  paths  of  three  different  grains  in  a 
suspension  of  mastic  are  shown  in  Fig.  74,  the  dots  representing 
the  successive  positions  occupied  by  the  particles  after  intervals  of 


COLLOIDS 


237 


30  seconds.  The  straight  line  joining  the  initial  and  final  posi- 
tions of  a  particle  is  called  the  horizontal  displacement  A,  of  the 
particle. 

If  the  time  taken  by  the 
particle  to  move  from  its 
initial  to  its  final  position 
be  t,  Einstein*  has  shown 
that  the  mean  value  of  the 
square  of  the  horizontal  dis- 
placement of  a  spherical  par- 
ticle of  radius  r,  ought  to 
be 

n  m        i 

(8) 


N 

where  f]  is  the  viscosity  of 
the  suspending  medium,  and 
where  the  other  symbols  have 
their  usual  significance. 

This    equation    was    tested  Flg'  74 

by   Perrin,   using    suspensions  of   gamboge    and   mastic.     Some 
of  the  results  obtained  are   given  in  the  following  table:  — 

VALUES  OF  AT  CALCULATED  BY  EINSTEIN'S  EQUATION 


Suspension. 

r  in 
microns.* 

mXlQi5. 

No.  of  dis- 
placements. 

A^X  10-22. 

Gamboge  in  water.  .  .  . 

0.367 

246 

1500 

69 

Gamboge     in     10%     solution     of 
glycerine  

0.385 

290 

100 

64 

Mastic  in  water  

0.52 

650 

1000 

73 

Mastic  in  27%  solution  of  urea  — 

5.50  - 

750,000 

100 

78 

*  The  micron  is  one-millionth  of  a  meter  or  one  ten-thousandth  of  a  centimeter. 

It  will  be  seen,  that  notwithstanding  the  large  variations  in  the 
granular  masses  of  the  different  suspensions  recorded  in  the  table, 
the  values  of  Nt  calculated  by  means  of  Einstein's  equation,  are 
quite  concordant.  Perrin  gives  as  the  mean  value  of  all  of  his 
experiments,  N  =  68.5  X  1022. 

Recent  Investigations  of  the  Brownian  Movement.  Nord- 
lundf  has  recently  repeated  Perrin's  experiments,  employing  a 
colloidal  solution  of  mercury,  and  an  arrangement  of  apparatus, 

*  Zeit.  Elektrochem.,  14,  235  (1908). 

t  Zeit.  phys.  Chem.,  87,  60  (1914). 


238 


THEORETICAL  CHEMISTRY 


whereby  the  movements  of  the  particles  could  be  recorded  photo- 
graphically. The  mean  value  of  N,  derived  from  twelve  carefully 
executed  experiments  was,  59  X  1022,  the  average  deviation  of  the 
results  of  the  individual  experiments  from  the  mean  being  approxi- 
mately 10  per  cent. 

The  Brownian  Movement  in  gases  has  been  studied  by  Milli- 
kan,*  and  also  by  Fletcher,  f  employing  a  minute  drop  of  oil  as  the 
suspended  particle.  In  the  gaseous  state,  where  the  intermolecular 
distances  are  greater  than  in  the  liquid  state,  not  only  are  the  colli- 
sions less  frequent,  but  the  mean  free  paths  are  appreciably  longer. 
These  conditions  are  favorable  to  the  study  of  the  Brownian  Move- 
ment, and  offer  an  opportunity  for  the  determination  of  the  Avo- 
gadro  Constant  with  a  high  degree  of  accuracy.  As  the  mean  of 
nearly  six  thousand  measurements,  Fletcher  gives  N  =  60.3  X 
1022,  this  value  being  accurate  to  within  1.2  per  cent. 

Density  of  Colloidal  Solutions.  As  we  have  seen,  suspensoids 
are  commonly  regarded  as  sols  in  which  the  disperse  phase  is  solid, 
while  emulsoids  are  considered  to  be  sols  in  which  the  disperse 
phase  is  liquid.  While  this  distinction  between  the  two  classes  of 
sols  is  generally  well  defined,  it  should  be  borne  in  mind  that  there 
are  colloidal  solutions  in  which  it  is  extremely  difficult  to  deter- 
mine the  physical  state  of  the  disperse  phase. 

The  fundamental  difference  between  suspensoids  and  emulsoids, 
manifests  itself  most  clearly  in  those  properties  which  undergo 
appreciable  change  in  consequence  of  solution.  Among  these,  may 
be  mentioned  density,  viscosity  and  surface  tension. 

It  was  shown  by  Linder  and  Picton,  J  that  the  density  of  sus- 
pensoids follows  the  law  of  mixtures.  This  is  clearly  shown  by 
the  following  table,  in  which  are  given  the  observed  and  calculated 
values  of  the  density  of  a  series  of  arsenious  sulphide  sols. 

DENSITY  OF  ARSENIOUS  SULPHIDE  SOLS 


As2S3  (grams  per 
liter). 

Density  (obs.). 

Density  (calc.). 

44 

1.033810 

1.033810 

22 

1.016880 

1.016905 

11 

1.008435 

1.008440 

2.45 

1.002110 

1.002100 

0.1719 

1.000137 

1.000134 

*  Phys.  Rev.,  i.  220  (1913).  f  Ibid.,  4,  453  (1914). 

$  Jour.  Chem.  Soc.,  67,  71  (1895). 


COLLOIDS 


239 


The  density  of  emulsoids,  on  the  other  hand,  cannot  be  calcu- 
lated from  the  composition  of  the  sol.  This  fact  may  be  taken  as 
evidence  in  favor  of  the  view,  that  a  closer  relation  exists  between 
the  disperse  phase  and  the  dispersion  medium  in  emulsoids  than  in 
suspensoids. 

Viscosity  of  Colloidal  Solutions.  Owing  to  the  fact  that  the 
concentrations  of  most  suspensoids  are  relatively  small,  it  follows 
that  their  viscosities  differ  but  little  from  the  viscosity  of  the  pure 
dispersion  medium.  In  general,  it  may  be  said  that  the  viscosity 
of  suspensoid  sols  is  slightly  greater  than  that  of  the  dispersion 
medium. 

On  the  other  hand,  the  viscosity  of  emulsoid  sols  is  frequently 
much  greater  than  that  of  the  pure  dispersion  medium.  The  vis- 
cosity of  emulsoids  also  increases  with  increasing  concentration, 
as  is  shown  by  the  data  of  the  following  table. 

VISCOSITY  OF  EMULSOIDS 


Sol. 

Temp.  20° 
Concentration. 

Viscosity. 

Gelatine  

Per  cent 
1 

0  021 

Gelatine  
Silicic  acid.  .  .  . 
Silicic  acid.  .  .  . 
Silicic  acid.  .  .  . 
Silicic  acid.  .  .  . 

2 
0.81 
0.99 
1.96 
3.67 

0.037 
0.012 
0.016 
0.032 
0.165 

Viscosity  of  water  at  20°  =  0.0120. 

Surface  Tension  of  Colloidal  Solutions.  The  surface  tension 
of  suspensoid  sols  has  been  shown  by  Linder  and  Picton  to  be 
practically  identical  with  that  of  the  dispersion  medium. 

As  a  general  rule,  the  surface  tension  of  emulsoid  sols  is  appre- 
ciably smaller  than  that  of  the  dispersion  medium.  According  to 
Quincke,*  the  surface  tensions  of  gelatine  sols  are  appreciably  less 
than  the  surface  tension  of  the  dispersion  medium.  The  differ- 
ence between  suspensoids  and  emulsoids,  in  respect  to  surface 
tension,  undoubtedly  accounts  for  the  fact,  that  in  general,  the 
former  are  not  adsorbed,  while  the  latter  are. 

Osmotic  Pressure  of  Colloidal  Solutions.  The  osmotic  pres- 
sure of  colloidal  solutions  is  very  small.  This  is  what  we  should 

*  Wied.  Ann.,  Ill,  35,  582  (1885). 


240  THEORETICAL  CHEMISTRY 

expect  with  solutions  of  substances  which  exhibit  a  slow  rate  of 
diffusion.  As  has  been  pointed  out,  diffusion  is  closely  connected 
with  osmotic  pressure;  hence,  if  the  rate  of  diffusion  is  slow,  the 
osmotic  pressure  exerted  by  the  solution  should  be  small.  In 
some  cases,  the  osmotic  pressure  is  so  small  as  to  escape  detection. 
The  experimental  determination  of  the  osmotic  pressure  of  col- 
loidal solutions  is  complicated  by  the  difficulty  of  removing  the 
last  traces  of  electrolytes  from  the  colloid.  Owing  to  their  great 
osmotic  activity,  the  presence  of  the  merest  trace  of  an  electrolyte 
may  mask  the  true  osmotic  effect  of  a  colloid.  It  should  be 
borne  in  mind,  however,  that  semipermeable  membranes  are  much 
less  permeable  by  colloids  than  by  electrolytes  and.  in  consequence 
of  this  fact,  the  impurities  in  the  colloid  would  be  gradually  re- 
moved by  prolonged  dialysis.  If  the  total  osmotic  pressure  were 
due  to  the  presence  of  small  amounts  of  impurities  in  the  colloid, 
then  as  these  are  removed,  the  pressure  should  steadily  diminish 
and  ultimately  become  zero.  As  a  matter  of  fact,  the  final  value 
of  the  osmotic  pressure  of  a  colloidal  solution,  although  generally 
very  small  is  never  zero.  This  final,  positive  value  of  the  osmotic 
pressure  has  been  shown  to  be  wholly  independent  of  the  method 
of  preparation  of  the  sol.  Although  differences  in  the  method  of 
preparation  may  introduce  different  impurities  which  give  rise  to 
different  initial  values  of  the  osmotic  pressure,  in  each  case  the 
same  final  value  si  obtained.  Of  course  it  must  be  admitted, 
that  the  possibility  exists  that  a  minute  portion  of  electrolyte 
which  cannot  be  removed  by  dialysis  is  retained  by  the  col- 
loid, but  even  then  it  is  difficult  to  account  for  the  constancy 
of  the  final  value  of  the  osmotic  pressure,  irrespective  of  the 
method  of  preparation  of  the  sol.  The  values  of  the  osmotic 
pressure  of  suspensoids  are  invariably  small,  and  by  no  means 
concordant. 

The  following  table  gives  the  results  obtained  by  Duclaux*  with 
colloidal  solutions  of  ferric  hydroxide. 

Inspection  of  he  table  shows  that,  even  in  the  most  concen- 
trated solution,  the  osmotic  pressure  is  very  small.  Furthermore, 
it  is  apparent  that,  although  the  osmotic  pressure  increases  with 
the  concentration,  the  variables  are  not  proportional.  Observa- 
tions on  the  variation  of  the  osmotic  pressure  of  colloidal  solu- 
tions with  temperature,  have  shown  that,  in  general,  as  the  tem- 
*  Compt.  rend.,  140,  1544  (1905);  Jour.  Chim.  Phys.,  7,  405  (1909). 


COLLOIDS 


241 


perature  is  raised  the  pressure  increases  at  a  more  rapid  rate 
than  that  required  by  the  law  of  Gay-Lussac. 

OSMOTIC  PRESSURE  OF  COLLOIDAL  FERRIC 
HYDROXIDE  SOLS.   AT  25° 


(First  Series) 

Concentration,  c. 

Pressure,  P. 

P/c 

Per  cent 

Cm.  of  Water 

0.15 

0.20 

1.4 

0.2 

0.55 

2.7 

0.4 

2.0 

5.0 

0.8 

7.0 

8.7 

1.84 

22.0 

12.0 

(Second  Series) 

1.08 

0.8 

0.7 

2.04 

2.8 

1.4 

3.05 

5.6 

1.8 

5.35 

12.5 

2.3 

8.86 

22.6 

2.5 

Employing  membranes  of  collodion  and  parchment  paper, 
Lillie,*  and  others,  have  demonstrated  that  the  values  of  the  os- 
motic pressure  of  emulsoids  are,  in  general,  considerably  greater 
than  the  corresponding  values  obtained  with  suspensoids.  The 
osmotic  pressures  of  several  typical  emulsoids  are  given  in  the 
following  table. 

•v 

OSMOTIC  PRESSURES  OF  EMULSOIDS 


Sol. 

Concentration 
(grams  per  liter). 

Osmotic  Pressure 
(mm.  of  mercury). 

Egg  albumin  

12.5 

20 

Gelatine  

12.5 

6 

Starch  iodide  
Dextrin  

30 
10 

15 
165 

It  will  be  observed,  that  the  values  of  the  osmotic  pressure  in  the 
preceding  table  are  appreciably  greater  than  those  given  for  ferric 
hydroxide  hydrosols.     This  is  in  agreement  with  the  well-estab- 
lished fact,  that  emulsoids  diffuse  more  rapidly  than  suspensoids. 
*  Am.  Jour.  PhysioL,  20,  127  (1907). 


242  THEORETICAL  CHEMISTRY 

It  has  been  observed,  that  the  value  of  the  osmotic  pressure  of 
gelatine  solutions,  at  ordinary  temperatures,  can  be  increased  by 
maintaining  the  solutions  at  a  higher  temperature  for  a  short  time, 
and  then  cooling  to  the  initial  temperature.  After  standing  for 
several  days  at  the  original  temperature,  however,  the  osmotic 
pressure  of  the  solution  returns  to  its  former  value.  This  phenom- 
enon would  seem  to  indicate  that  the  osmotic  pressure  of  colloidal 
solutions  is  not  completely  defined  by  the  two  variables,  tempera- 
ture and  concentration.  It  has  been  suggested,  that  the  degree  of 
aggregation  of  the  colloid  is  partially  dependent  upon  the  tempera- 
ture, the  molecular  aggregates  tending  to  break  up  as  the  temper- 
ature is  raised,  thus  increasing  the  number  of  dissolved  units  and 
therefore  causing  a  corresponding  increase  in  the  osmotic  pressure. 

Molecular  Weight  of  Colloids.  We  have  already  learned,  that 
the  knowledge  of  the  osmotic  pressure  of  a  solution  enables  us  to 
calculate  the  molecular  weight  of  the  solute,  provided  the  solu- 
tion is  dilute,  and  obeys  the  gas  laws.  As  we  have  seen,  other 
factors  than  concentration  and  temperature  determine  the  osmotic 
pressure  -  of  colloidal  solutions,  so  that  we  are  not  justified 
in  attempting  to  calculate  the  molecular  weight  of  a  colloid  from 
the  observed  value  of  the  osmotic  pressure  of  its  solution.  Values 
for  the  molecular  weight  of  colloids,  calculated  from  their  effect 
on  the  vapor  pressure,  the  boiling-point,  and  the  freezing-point  of 
the  solvent  are  also  untrustworthy,  since  the  same  factors  which 
influence  the  osmotic  pressure,  necessarily  affect  these  related 
properties.  This  becomes  evident  when  we  reflect  that  an  osmotic 
pressure  of  1  mm.  of  mercury  corresponds  to  a  depression  of  the 
freezing-point  of  water  about  0°.0001.  Owing  to  the  difficulty  of 
obtaining  absolutely  pure  emulsoid  sols,  all  determinations  of  their 
freezing-point  depressions  must  be  affected  with  an  experimental 
error  appreciably  larger  than  the  observed  depression.  Hydrosols 
of  albumin,  gelatine,  etc.,  prepared  with  extreme  care  by  Bruni 
and  Pappada,*  failed  to  produce  any  detectable  depression  of  the 
freezing-point  of  water. 

Electroendosmosis.  The  movement  of  a  liquid  through  a 
porous  diaphragm,  due  to  the  passage  of  an  electric  current  be- 
tween two  electrodes  placed  on  opposite  sides  of  the  diaphragm,  is 
known  as  electroendosmosis.  This  phenomenon,  which  was  first 
observed  by  Reuss,  in  1807,  has  since  been  made  the  subject  of 

*  Rend.  R.  Accad.  dei  Lincei,  (5),  9,  354  (1900). 


COLLOIDS  243 

numerous  investigations  by  Wiedemann,*  Quinckef  and  Pen-in,! 
the  latter  having  worked  out  a  satisfactory  theoretical  interpreta- 
tion of  the  phenomenon.  If  a  porous  partition  be  placed  in  the 
horizontal  portion  of  a  U-tube,  and  an  electrode  be  inserted  in  each 
arm  of  the  tube,  it  will  be  found,  on  filling  the  tube  with  a  feebly 
conducting  liquid,  and  passing  a  current,  that  the  liquid  will  com- 
mence to  rise  in  one  arm  of  the  tube,  and  will  continue  to  rise,  until 
a  definite  equilibrium  is  established.  For  a  given  difference  of 
potential  between  the  two  electrodes,  there  will  be  a  definite  differ- 
ence in  the  level  of  the  liquid  in  the  two  arms  of  the  tube.  The 
majority  of  substances  acquire  a  negative  electric  charge  when 
immersed  in  w^ater.  The  water,  under  these  conditions,  becomes 
positively  charged  and  will,  in  consequence,  migrate  toward  the 
cathode.  On  the  other  hand,  certain  substances  acquire  a  posi- 
tive charge,  on  immersion  in  water,  and  in  these  cases  the  direction 
of  migration  will  obviously  be  reversed. 

It  has  been  found  that  acids  cause  negative  diaphragms  to  be- 
come less  negative,  and  positive  diaphragms  to  become  more  posi- 
tive. The  action  of  alkalies  is,  as  we  should  expect,  the  reverse  of 
that  of  acids.  There  is  an  interesting  connection  between  the 
valence  of  the  ions,  resulting  from  the  dissociation  of  dissolved 
salts,  and  the  difference  of  potential  existing  between  the  liquid 
and  the  diaphragm.  When  the  diaphragm  is  positively  charged, 
the  difference  of  potential  is  found  to  be  conditioned  by  the 
valence  of  the  anion,  and  when  the  diaphragm  is  negatively 
charged,  the  difference  of  potential  is  determined  by  the  valence 
of  the  cation. 

Cataphoresis.  When  a  difference  of  potential  is  established 
between  two  electrodes  immersed  in  a  suspension  of  finely-divided 
quartz,  or  shellac,  the  suspended  particles  move  toward  the  positive 
electrode.  This  phenomenon  is  called  cataphoresis,  and  was  first 
observed  by  Linder  and  Picton.§  They  showed  that  when  the 
terminals  of  an  electric  battery  are  connected  to  two  platinum 
electrodes  dipping  into  a  colloidal  solution  of  arsenious  sulphide, 
there  is  a  gradual  migration  of  the  colloid  to  the  positive  pole.  A 
similar  experiment  with  a  solution  of  colloidal  ferric  hydroxide, 

*  Pogg.  Ann.,  87,  321  (1852). 
t  Ibid.,  113,  513  (1861). 
J  Jour.  Chim.  Phys.,  2,  601  (1904). 
§  Jour.  Chem.  Soc.,  61,  148  (1892). 


244 


THEORETICAL  CHEMISTRY 


resulted  in  the  transport  of  the  dissolved  colloid  to  the  negative 
pole.  It  follows,  therefore,  that  the  particles  of  colloidal  arsenious 
sulphide  are  negatively  charged,  while  those  of  colloidal  ferric 
hydroxide  carry  a  positive  charge.  It  has  been  found,  that  most 
colloids  carry  an  electric  charge.  In  the  following  table  some 
typical  colloids  are  classified  according  to  the  character  of  their 
electrification  in  aqueous  solution. 

ELECTRICAL  CHARGES  OF  HYDROSOLS 


Electro-positive. 

Electro-negative. 

Metallic  hydroxides 
Methyl  violet 
Methylene  blue 
Magdala  red 
Bismarck  brown 
Ifemoglobin 

All  the  metals 
Metallic  sulphides 
Aniline  blue 
Indigo 
Eosine 
Starch 

The  nature  of  the  charge  varies  with  the  dispersion  medium  used; 
colloidal  solutions  in  turpentine,  for  example,  having  charges 
opposite  to  those  in  water. 

Direct  measurements  of  the  velocity  with  which  the  particles 
move  in  cataphoresis  have  been  made  by  Cotton  and  Mouton.* 
By  observing  with  a  microscope  the  distance  over  which  a  single 
particle  traveled  in  a  given  interval  of  time,  under  a  definite  po- 
tential gradient,  they  calculated  the  average  velocity  of  migration 
of  a  number  of  suspensoids.  The  following  table  gives  the  vel- 
ocity of  migration  of  a  few  typical  suspensoids. 

VELOCITY  OF  MIGRATION  OF  SUSPENSOIDS  f 


Suspensoid. 

Average  Diameter 
of  Particles. 

Velocity  cm.  /sec.  for 
Unit  Potential 
Gradient. 

Arsenic  trisulphide 

50  M 

22X10~6 

8uartz  .  .                    ... 

30X10~5 

old  (colloidal)     

<100  /JLfJL 

40X10"5 

Platinum  (colloidal)  

<  100  MM 

30X10^5 

Silver  (colloidal)  

<  100  MM 

23.6X10"5 

Bismuth  (colloidal) 

<100  MM 

11  OX10~5 

Lead  (colloidal) 

<  100  MM 

12  OX10~5 

Iron  (colloidal)  .  . 

<  100  MM 

19  OX10"5 

Ferric  hydroxide  (colloidal)  .  . 

100  MM 

30  OX10~5 

t  Freundlich,  Kapillarchemie,  p.  234. 

*  Jour.  Chim.  Phys.,  4,  365  (1906). 


COLLOIDS  245 

It  will  be  observed  that  not  only  are  the  velocities  of  migration 
nearly  constant,  but  also  that  they  are  apparently  independent  of 
the  size  and  nature  of  the  particles. 

The  presence  of  electrolytes,  especially  acids  and  bases,  exer- 
cises a  marked  effect  upon  the  electrical  behavior  of  suspensoids. 
Owing  to  the  comparative  instability  of  suspensoids,  the  addi- 
tion of  electrolytes  usually  results  in  the  complete  precipitation  of 
the  colloid. 

Emulsoids  also  show  the  phenomenon  of  cataphoresis,  but  their 
velocities  of  migration  are  appreciably  less  than  the  corresponding 
velocities  of  suspensoids,  and  their  behavior  in  an  electric  field  is 
such  as  to  make  it  appear  quite  probable  that  the  character  of 
their  electric  charge  is  entirely  fortuitous.  Furthermore,  emul- 
soids  are  much  more  susceptible  to  the  influence  of  electrolytes 
than  are  suspensoids. 

W.  B.  Hardy*  has  found  that  the  direction  of  migration  of 
albumin,  modified  by  heating  to  100°  C.,  is  dependent  upon  the 
reaction  of  the  dispersion  medium.  A  very  small  quantity  of  free 
base  causes  the  particles  of  albumin  to  move  toward  the  positive 
electrode,  while  the  addition  of  an  equally  small  amount  of  acid 
results  in  a  reversal  of  the  direction  of  migration.  Similar  rever- 
sals of  charge  have  been  observed  by  Burton  f  in  colloidal  solutions 
of  gold  and  silver.  When  small  amounts  of  aluminium  sulphate 
are  added  to  colloidal  solutions  of  these  metals,  the  charge  is 
gradually  neutralized  and,  eventually,  the  colloidal  particles  ac- 
quire a  reversed  charge. 

Electrical  Conductance  of  Colloidal  Solutions.  The  electrical 
conductance  of  suspensoids  differs  so  slightly  from  that  of  the  pure 
dispersion  medium,  that  it  is  difficult  to  decide  whether  the  small 
increase  in  conductance  may  not  be  due  to  the  presence  of  traces 
of  adsorbed  electrolytes.  In  order  to  ascertain  to  what  extent  the 
conductance  of  suspensoids  is  dependent  upon  the  presence  of 
adsorbed  electrolytes,  Whitney  and  Blake  J  investigated  the  effect 
of  successive  electrolyses  upon  the  conductance  of  a  gold  hydrosol. 
If  the  pure  sol  is  incapable  of  enhancing  the  conductance  of  the 
dispersion  medium,  then  as  the  sol  is  subjected  to  successive  elec- 
trolyses, the  conductance  should  steadily  diminish  and,  ultimately, 
become  identical  with  that  of  the  dispersion  medium.  Whitney 

*  Jour.  Physiol.,  24,  288  (1899).  t  Phil-  Mag.,  12,  472  (1906). 

J  Jour.  Am.  Chem.  Soc.,  26,  1339  (1904). 


246  THEORETICAL  CHEMISTRY 

and  Blake  found  that  the  conductance  converged  to  a  definite 
limiting  value,  which  was  slightly  greater  than  the  conductance  of 
the  dispersion  medium.  From  these  experiments  we  seem  to  be 
warranted  in  concluding,  that  suspensoids  conduct  the  electric 
current  very  feebly. 

Emulsoids  appear  to  conduct  rather  better  than  suspensoids.* 
Whitney  and  Blake  measured  the  conductance  of  silicic  acid  and 
gelatine  sols,  and  found  the  specific  conductance  of  the  former  to  be 
100  X  10~6  reciprocal  ohms,  and  that  of  the  latter  to  be  68  X  10"6 
reciprocal  ohms.  On  the  other  hand,  Paulif  found  that  an  al- 
bumin sol,  which  had  been  prepared  with  extreme  care,  was  vir- 
tually a  non-conductor.  It  should  be  remembered,  however,  that 
the  albumins  are  closely  related  to  the  simple  amino-acids,  which 
are  known  to  be  exceedingly  poor  conductors. 

Precipitation  of  Colloids  by  Electrolytes.  One  of  the  most 
important  and  interesting  divisions  of  the  chemistry  of  colloids,  is 
that  which  treats  of  the  precipitation  of  suspensoids  and  emulsoids 
by  electrolytes.  In  general,  it  may  be  said  that  the  precipitation 
of  colloids  by  electrolytes  is  an  irreversible  process.  Colloidal 
solutions  are  more  or  less  unstable  systems,  irrespective  of  the 
methods  employed  in  their  preparation,  and  the  addition  of  a 
small  amount  of  an  electrolyte  is  usually  found  to  be  sufficient  to 
cause  the  sol  immediately  to  become  opalescent,  and  ultimately 
to  precipitate,  leaving  the  dispersion  medium  perfectly  clear  and 
free  from  the  disperse  phase. 

Some  exceptions  to  this  general  statement  as  to  the  behavior 
of  colloids  are  known.  For  example,  Whitney  and  Blake  | 
found  that  precipitated  gold  could  be  caused  to  return  to  the 
colloidal  state  by  treatment  with  ammonia,  while  Linder  and 
Picton§  discovered  that  a  ferric  hydroxide  hydrosol,  which  had 
been  precipitated  with  sodium  chloride,  could  be  restored  to 
the  colloidal  condition  by  simply  removing  the  electrolyte  with 
water.  The  sedimentation  of  suspensions,  such  as  kaolin  in 
water,  is  also  promoted  by  the  addition  of  electrolytes.  On  the 
other  hand,  the  addition  of  some  non-electrolytes  frequently 
causes  an  increase  in  the  stability  of  a  suspensoid. 

*  This  may  be  due  to  greater  difficulties  involved  in  purification. 

t  Beitrag.  Chem.  Phys.  Path.,  7,  531  (1906). 

|  Jour.  Am.  Chem.  Soc.,  26,  1341  (1904). 

§  Jour.  Chem.  Soc.;  61,  114  (1892);  87, 1924  (1905). 


COLLOIDS 


247 


Precipitation  of  Suspensoids.  The  phenomenon  of  the  pre- 
cipitation of  suspensoids  has  been  carefully  investigated  by 
Freundlich.*  He  has  found,  that  an  amount  of  electrolyte  which 
is  incapable  of  bringing  about  an  instantaneous  precipitation,  may 
become  effective  after  an  interval  of  time.  He  has  also  shown, 
that  the  total  quantity  of  electrolyte  required  to  precipitate  a 
suspensoid  completely,  depends  upon  whether  the  electrolyte  is 
added  all  at  one  time,  or  in  successive  portions. 

In  order  to  compare  the  precipitating  action  of  various  electro- 
lytes, Freundlich  proposed  the  following  procedure,  which  pre- 
vents the  possibility  of  irregularities  due  to  the  time  factor:  — 
To  20  cc.  of  a  solution  of  a  suspensoid.  2  cc.  of  the  solution  of 
the  electrolyte  are  added,  the  resulting  solution  being  shaken 
vigorously;  the  mixture  is  then  set  aside  for  two  hours,  after 
which  a  small  portion  is  filtered  off,  and  the  filtrate  is  examined 
for  the  suspensoid.  In  the  following  table,  some  of  the  results 
obtained  by  Freundlich  with  colloidal  solutions  of  ferric  hy- 
droxide are  given.  The  data  represent  the  minimum  concen- 
tration for  each  electrolyte  which  produced  precipitation  in  two 
hours. 

PRECIPITATING  ACTION   OF  ELECTROLYTES  ON  FERRIC 
HYDROXIDE  HYDROSOL 

(16  milli-mols  Fe(OH)3  per  liter) 


Concentration 

Concentration 

Electrolyte 

(milli-mols  per 

Electrolyte 

(milli-mols  per 

liter) 

liter) 

NaCl 

9  25 

HC1 

400  ca. 

KC1  

9.03 

Ba(OH)2  

0.42 

BaCl2  

9.64 

KNO3 

11  9 

KoSO4.  . 

0.204 

KBr  

12.5 

MgS04  

0.217 

Ba(NO3)2 

14  0 

K2Cr2O7  

0.194 

KI  

16.2 

H,SO4  

0.5  ca. 

It  will  be  seen  that  very  small  amounts  of  the  electrolytes  are 
required  to  precipitate  the  suspensoid,  and  further,  that  the  pre- 
cipitating power  of  an  electrolyte  is  dependent  upon  the  charge  of 
the  negative  ion.  The  greater  the  charge,  the  smaller  is  the 
quantity  of  electrolyte  required  to  produce  precipitation. 

The  significance  of  the  relation  between  ionic  charge  and  pre- 
*  Zeit.  phys.  Chem.,  44,  131  (1903). 


248 


THEORETICAL  CHEMISTRY 


cipitating  power  was  first  pointed  out  by  Hardy,*  who  formulated 
the  following  rule :  —  The  precipitation  of  a  colloidal  solution  is  de- 
termined by  that  ion  of  an  added  electrolyte  which  has  an  electric 
charge  opposite  in  sign  to  that  of  the  colloidal  particles. 

It  has  already  been  pointed  out,  that  colloidal  particles  of  arse- 
nious  sulphiqle  are  negatively  charged,  hence,  according  to 
Hardy's  rule,  the  positive  ions  of  the  added  electrolyte  will  condi- 
tion the  precipitation  of  the  suspensoid.  The  experiments  of 
Freundlich  confirm  this  prediction,  as  is  shown  by  the  following 
table: 

PRECIPITATING  ACTION  OF  ELECTROLYTES  ON  ARSENIOUS 
SULPHIDE  HYDROSOL 

(7.54  milli-mols  As2S3  per  liter) 


Electrolyte 

Concentration 
(milli-mols  per 
liter) 

Electrolyte 

Concentration 
(milli-mols  per 
liter) 

KC1.. 
KN03  
KC2H3Oa  

49.5 

50.0 
110.0 

Cadi.. 

SrCU...:  
BaCl2  

0.649 
0.635 
0.691 

NaCl  

51.0 

Ba(NO3)2  

0.687 

LiCl 

58.4 

ZnCl2 

0  685 

MgCl2 

0.717 

A1C13  . 

0  093 

MgSO4 

0.810 

A1(NO3)3  

0  095 

Precipitation  and  Valence.  An  examination  of  the  preceding 
tables  reveals  the  fact,  that  although  the  ionic  concentration  neces- 
sary to  bring  about  precipitation,  in  accordance  with  Hardy's 
rule,  decreases  with  increasing  valence,  the  diminution  in  con- 
centration is  not,  as  we  might  expect,  inversely  proportional  to 
the  valence  of  the  precipitating  ion.  The  absence  of  any  simple 
quantitative  relation  between  the  valence  of  an  ion  and  its  pre- 
cipitating concentration,  is  undoubtedly  due  to  the  influence  of 
several  potent  factors,  such  as  adsorption,  and  the  protective  ac- 
tion of  ions  whose  electric  charge  is  of  the  same  sign  as  that  of  the 
colloidal  substance. 

Precipitation   of   Emulsoids.     The   action   of   electrolytes   on 

emulsoids  is  much  more  obscure  than  the  action  of  electrolytes  on 

suspensoids.      Nothing  approaching  a  generalization  similar  to 

Hardy's  rule  for  the  precipitation  of  suspensoids,  has  been  found 

*  Zeit.  phys.  Chem.,  33,  385  (1900). 


COLLOIDS  249 

to  apply  to  the  precipitation  phenomena  manifested  by  emulsoids. 
Owing  to  the  fact  that  emulsoids  are  liquids,  and  in  consequence 
of  their  greater  degree  of  dispersion,  it  has  been  suggested  that 
emulsoids  probably  resemble  true  solutions  more  closely  than  sus- 
pensoids.  In  fact,  there  is  reason  for  assuming  that  a  portion  of 
the  colloid  is  actually  dissolved  in  the  dispersion  medium.  This 
may  account  for  the  fact,  that  the  precipitation  of  emulsoids  is 
sometimes  reversible,  and  sometimes  irreversible.  It  is  to  be  re- 
gretted that,  up  to  the  present  time,  so  many  of  the  investigations 
on  emulsoids  have  been  carried  out  with  materials  of  questionable 
purity  and  of  insufficient  uniformity. 

The  addition  of  salts  to  gelatine  generally  causes  irreversible 
precipitation,  provided  the  concentration  of  the  salt  is  not  too  low. 
The  precipitation  of  albumin  by  some  salts  is  reversible,  while 
by  others  it  is  irreversible.  Those  transformations  which  are 
initially  reversible,  gradually  become  irreversible  on  standing. 
Although  relatively  concentrated  solutions  of  salts  of  the  alkalies 
and  alkaline  earths  are  required  to  precipitate  albumin,  very  dilute 
solutions  of  the  salts  of  the  heavy  metals  are  found  to  be  sufficient 
to  bring  about  complete  precipitation. 

Action  of  Heat  on  Emulsoids.  When  an  albumin  hydrosol  is 
gradually  heated,  a  temperature  is  ultimately  reached  at  which 
coagulation  occurs.  The  exact  nature  of  this  transformation  is 
not  understood,  but  it  is  believed  to  be  largely  chemical.  This 
belief  is  based  upon  the  fact,  that  the  reaction  of  the  natural  albu- 
mins toward  litmus  is  altered  by  heating.  Slight  acidity  of  the 
sol  is  essential  to  complete  coagulation,  while  an  excess  or  a  de- 
ficiency of  acids,  causes  a  portion  of  the  albumin  to  remain  in  the 
sol.  The  presence  of  various  salts  has  been  found  to  exert  a 
marked  influence  on  the  temperature  of  coagulation  of  albumin. 
The  coagulation  temperature  is  invariably  raised  at  first,  attaining 
a  constant  value  in  some  cases,  while  in  others  it  decreases,  after 
reaching  a  maximum  temperature. 

The  effect  of  heat  on  a  gelatine  sol  is  very  different  from  its 
effect  on  an  albumin  sol.  If  a  fairly  concentrated  gelatine  sol  is 
heated  and  then  permitted  to  cool,  it  sets  into  a  jelly  which  is  not 
reconverted  into  a  sol  when  the  temperature  is  again  raised.  Fur- 
thermore, the  change  does  not  take  place  at  a  definite  temperature. 

In  studying  the  phenomenon  of  gelation,  either  the  temperature 
or  the  time  of  gelation  may  be  determined.  The  melting-point 


250  THEORETICAL  CHEMISTRY 

of  pure  gelatine  ranges  from  26°  to  29°,  while  the  solidifying  tem- 
perature lies  between  25°  and  18°.*  The  melting  and  solidifying 
temperature  of  gelatine  sols  vary  with  the  concentration;  a  5  per 
cent  sol  melts  at  26°.  1  and  solidifies  at  17°.8,  while  a  15  per  cent 
sol  melts  at  29°.4  and  solidifies  at  25°. 5.  The  temperature  of  the 
gel-sol  transformation  is  affected  by  the  addition  of  salts,  some 
tending  to  raise  the  temperature  of  gelation,  and  others  to  lower 
it.  The  order  of  the  anions  arranged  according  to  their  influence 
on  the  gel-sol  transformation  is  as  follows :  — 
Raising  temperature :  S04  >  Citrate  >  Tartrate  >  Acetate  (H20) . 
Lowering  temperature:  (H2O)  Cl  <  C1O3  <  NO3  <  Br  <  I. 

The  same  order  was  found  by  Schroeder,f  in  an  investigation  of  the 
viscosity  of  gelatine  sols. 

It  is  noteworthy,  that  the  influence  of  salts  on  the  temperature 
of  gelation  of  agar-agar,  and  other  similar  substances,  is  analogous 
to  their  effect  on  gelatine,  the  same  lytropic  sequence  being  main- 
tained. 

Protective  Colloids.  The  precipitating  action  of  electrolytes 
on  suspensoids  may  be  inhibited  by  adding  to  the  solution  of  the 
suspensoid,  a  reversible  colloid.  The  protective  action  of  a  re- 
versible colloid  is  not  due,  as  might  be  supposed,  to  the  increased 
viscosity  of  the  medium  and  the  resultant  resistance  to  sedimenta- 
tion, since  amounts  of  a  reversible  colloid,  too  minute  to  produce 
any  appreciable  increase  in  the  viscosity  of  the  medium,  can  pre- 
vent precipitation.  Thus,  BechholdJ  has  shown,  that  while  a 
mixture  of  1  cc.  of  a  suspension  of  mastic  and  1  cc.  of  a  0.1  molar 
solution  of  MgSO4,  diluted  to  3  cc.  with  water,  is  completely  pre- 
cipitated in  15  minutes,  no  precipitation  will  occur  within  24  hours, 
if  two  drops  of  a  1  per  cent  solution  of  gelatine  be  added,  before 
diluting  to  3  cc.  Gum  arabic  and  ox-blood  serum  exert  a  similar 
protective  action  when  added  to  a  suspension  of  mastic. 

The  protective  power  of  reversible  colloids  differs  widely, 
and  Zsigmondy§  has  attempted  to  make  this  the  basis  of  a 
method  of  classification  of  colloidal  substances.  A  red  colloidal 
gold  sol  becomes  blue  on  the  addition  of  a  small  amount  of 
sodium  chloride,  owing  to  the  increase  in  the  size  of  the  colloidal 

*  See  Sheppard  &  Sweet.    Jour.  Ind.  and  Eng.  Chem.  13,  423  (1921). 
f  Zeit.  phys.  Chem.,  45,  75  (1903). 
t  Zeit.  phys.  Chem.,  48,  408  (1904). 
§  Zeit.  analyt.  Chem.,  40,  697  (1901). 


COLLOIDS 


251 


particles.  Various  colloidal  substances  when  added  to  a  red 
colored  gold  sol,  protect  the  colloidal  particles  from  precipitation 
by  a  solution  of  sodium  chloride,  no  change  in  color  following  the 
addition  of  the  electrolyte.  A  definite  amount  of  each  colloidal- 
substance  is  required  to  prevent  the  change,  from  red  to  blue,  in 
the  color  of  the  gold  sol.  In  employing  this  color  change  as  a 
means  of  differentiating  colloidal  substances,  Zsigmondy  intro- 
duced the  "  gold  number."  This  may  be  denned,  as  the  weight  in 
milligrams  of  a  colloidal  substance  which  is  just  insufficient  to 
prevent  the  change  from  red  to  blue  in  10  cc.  of  a  gold  sol,  after 
the  addition  of  1  cc.  of  a  10  per  cent  solution  of  sodium  chloride. 
The  following  table  gives  the  gold  numbers  of  a  few  colloids. 

GOLD  NUMBERS  OF  COLLOIDS 


Colloid 

Gold  Number 

Gelatine 

0  005-0  01* 

Casein  (in  ammonia)  

0.01 

Egg-albumin  

0.15-0.25 

Gum  arabic 

0  15-0  25'  0  5-4 

Dextrin  

6-12;  10-20 

Starch,  wheat 

4-6  (about) 

Starch,  potato  

25  (about) 

Sodium  stearate 

10  (at  60°);  001  (at  100°) 

Sodium  oleate  ...    . 

0  4-1 

Cane  sugar  

8 

Urea  

8 

*  See  Elliott  and  Sheppard.      Jour.  Ind.  and  Eng.  Chem.  13,  699  (1921). 

The  gold  number  has  proven  useful  in  differentiating  the  various 
kinds  of  albumin,  as  is  shown  in  the  following  table. 

GOLD  NUMBERS  OF  ALBUMINS 


Albumin. 

Gold  Number. 

Egg  white  (fresh)  

0.08 

Globulin 

0  02-0  05 

Ovomucoid 

0  04-0  08 

Albumin  (Merck)  

0.1-0.3 

Albumin  (cryst.)  

2-8 

Albumin  (alkaline)  

0.006-0.04 

The  addition  of  alkali  to  any  one  of  the  first  five  albumins  of  the 
above  table  reduces  the  gold  number  to  that  of  alkaline  albumin. 
Sulphide  sols  may  be  protected  as  well  as  metallic  sols,  and  further- 


252 


THEORETICAL  CHEMISTRY 


more,  the  ability  to  exert  protective  action  is  not.  confined  to  or- 
ganic colloids  alone. 

In  general,  it  may  be  said  that  when  a  suspensoid  sol  is  mixed 
with  an  emulsoid  sol  in  the  proper  proportions,  the  suspensoid  sol 
acquires  most  of  the  characteristic  properties  of  the  protecting 
colloid.  The  masking  of  the  properties  of  a  suspensoid  sol  by  a 
protecting  colloid,  is  probably  to  be  ascribed  to  the  formation  of  a 
thin  film  of  adsorbed  emulsoid  over  the  suspensoid. 

Mutual  Precipitation.  A  further  deduction  from  the  elec- 
trical theory  of  precipitation  is,  that  when  two  oppositely-charged 
colloids  are  mixed,  they  should  precipitate  each  other,  and  the 
resulting  precipitate  should  contain  both  colloids.  Experiments 
carried  out  by  Biltz*  have  confirmed  these  predictions.  He 
showed  that  when  a  solution  of  a  positively-charged  colloid  is 
added  to  a  solution  of  a  negatively-charged  colloid,  precipitation 
occurs,  unless  the  quantity  of  the  added  colloid  is  either  relatively 
very  large  or  very  small.  He  also  showed,  that  when  two  colloids 
of  the  same  electrical  sign  are  mixed,  no  precipitation  occurs. 
Just  as  the  amount  of  precipitation  caused  by  the  addition  of  an 
electrolyte  to  a  sol  is  conditioned  by  the  rate  at  which  the  electro- 
lyte is  added,  so  also  the  precipitation  of  one  colloid  by  another  is 
dependent  upon  the  manner  in  which  the  two  sols  are  mixed.  The 
extent  to  which  one  sol  is  precipitated  by  another  sol  of  opposite 
sign,  is  largely  determined  by  the  amount  of  the  one  that  is  added 
to  a  definite  amount  of  the  other.  This  is  clearly  shown  by 
the  following  table,  which  gives  the  results  obtained  by  Biltz 
on  adding  ferric  hydroxide  sol  to  2  cc.  of  an  antimony  trisulphide 
sol  containing  2.8  mg.  per  cc. 

PRECIPITATION  OF  COLLOIDAL  ANTIMONY  TRISULPHIDE 
BY  COLLOIDAL  FERRIC  HYDROXIDE 


Fe2O3(mg.). 

Immediate  Result. 

Result  After  One  Hour. 

0.8 

Cloudy 

Almost  homogeneous 

3.2 

Small  flakes 

Unchanged 

4.8 

Flakes 

Yellow  liquid 

6.4 
8.0 

Complete  precipitation 
Slow  precipitation 

Complete  precipitation 
Complete  precipitation 

12.8 

Cloudy 

Slight  precipitation 

20.8 

Cloudy 

Homogeneous 

Berichte,  37,  1095  (1904). 


COLLOIDS  253 

It  will  be  seen,  that  the  addition  of  a  small  amount  of  ferric 
hydroxide  produces  hardly  any  precipitation.  With  the  addition 
of  larger  amounts  of  ferric  hydroxide,  the  amount  of  precipitation 
increases,  until  finally,  complete  precipitation  is  attained.  The 
addition  of  larger  quantities  of  ferric  hydroxide  produces  either 
little,  or  no  precipitation.  It  has  been  found,  that  at  the  concen- 
tration which  just  produces  complete  precipitation,  the  electrical 
charges  of  the  two  sols  are  equivalent.  When  the  amount  of 
ferric  hydroxide  exceeds  that  required  for  complete  precipitation, 
it  is  more  than  probable  that  the  particles  of  colloidal  antimony 
trisulphide  are  completely  enveloped  by  the  particles  of  ferric  hy- 
droxide, and  thereby  rendered  inactive. 

When  we  come  to  study  the  action  of  one  emulsoid  on  another, 
we  find,  as  might  be  expected  from  the  general  behavior  of  emul- 
soids  toward  electrolytes,  that  the  phenomena  are  more  complex 
and  very  much  less  well-defined  than  with  suspensoids.  Although 
mutual  precipitation  does  take  place  with  emulsoids,  the  close 
resemblance  between  emulsoids  and  true  solutions  renders  the 
phenomenon  more  or  less  indistinct. 

When  a  suspensoid  sol  is  added  to  an  emulsoid  sol  having  an 
opposite  electric  charge,  precipitation  may  or  may  not  occur, 
according  to  the  relative  amounts  of  the  two  colloids  in  the  mix- 
ture. When  the  two  colloids  are  present  in  electrically  equivalent 
quantities,  either  precipitation  will  occur,  or  one  colloid  will  exert 
a  protective  action  on  the  other. 

Emulsions.  An_  emulsion  is  a  two-phase  system  in  which  both 
the_disperse  phase  and  the  dispersion  medium  are  liquids.  If  an 
emulsion  contains  relatively  little  of  the  disperse  phase,  its  prop- 
erties resemble  those  of  the  suspensoids.  That  is,  the  disperse 
phase  exhibits  the  Brownian  movement,  is  precipitated  by  elec- 
trolytes, and  can  even  be  separated  from  the  dispersion  medium 
by  the  process  of  ultrafiltration. 

Although  a  dilute  emulsion  differs  but  little  from  a  colloidal 
solution  of  the  suspensoid  type,  this  resemblance  rapidly  dis- 
appears as  the  concentration  of  the  emulsion  is  increased.  When 
the  concentration  of  an  emulsion  is  such  that  the  two  components 
are  present  in  nearly  equal  amounts,  both  phases  are  found  to  be 
in  a  highly  dispersed  condition.  One  phase,  termed  the  external 
phase,  exists  in  the  form  of  continuous  films,  the  surfaces  of  which 
are  convex  toward  the  phase  itself,  while  the  other  phase,  called 


254  THEORETICAL  CHEMISTRY 

the  internal  phase,  exists  in  the  form  of  minute  drops,  concave 
toward  that  phase.*  Thus,  in  emulsions  of  oil  in  water,  the  water 
is  the  external  phase,  while  the  oil,  which  is  present  in  the  form  of 
minute  drops,  constitutes  the  internal  phase. 

The  requisite  conditions  for  the  formation  of  a  stable  emulsion 
are,  (1)  that  the  size  of  the  drops  shall  be  such  that  they  will  re- 
main in  suspension,  and,  (2)  that  the  drops  shall  be  enveloped  in  a 
film  having  sufficient  strength  to  prevent  their  coalescence.  In 
order  to  insure  the  stability  of  an  emulsion,  it  is  necessary,  there- 
fore, to  add  some  substance  to  the  system  which  will  lower  the 
interfacial  tension  between  the  two  phases,  and  thus  produce  a 
film  possessing  the  desired  physical  properties.  Such  a  substance 
is  known  as  an  emulsifier.  Sodium  and  potassium  oleates  are 
commonly  used  as  emulsifiers,  while  certain  gums  find  similar 
application  in  the  preparation  of  various  pharmaceutical  emul- 
sions. If  the  sign  of  the  interfacial  tension  be  altered  by  changing 
the  emulsifier,  a  reversal  of  the  internal  and  external  phases  will 
occur.  Thus,  if  a  certain  emulsifier  lowers  the  surface  tension 
on  the  water-side  of  the  interface  more  than  it  does  on  the  oil- 
side,  in  a  mixture  of  oil  and  water,  the  interface  will  tend  to  be- 
come convex  on  the  water-side,  and  an  emulsion  of  oil  in  water 
will  result.  On  the  other  hand,  if  the  emulsifier  lowers  the  sur- 
face tension  on  the  oil-side  of  the  interface  more  than  it  does  on 
the  water-side,  the  curvature  of  the  interface  will  then  tend  to 
become  concave  on  the  water-side,  and  an  emulsion  of  water  in 
oil  will  be  formed.  For  example,  Clowes  f  has  shown  that  when 
olive  oil  is  shaken  with  a  dilute  solution  of  sodium  hydroxide,  an 
emulsion  of  oil  in  water  is  formed,  but  when  more  than  an  equiva- 
lent amount  of  calcium  chloride  is  added,  a  reversal  of  the  phases 
occurs,  and  an  emulsion  of  water  in  oil  results.  On  addition  of 
more  sodium  hydroxide  to  the  water-oil  emulsion,  it  reverts  to 
the  original  oil- water  emulsion. 

Emulsions  of  kerosene  in  water,  containing  as  high  as  99  per 
cent  of  oil  in  a  1  per  cent  soap  solution  have  been  prepared  and 
studied  by  Pickering.  J  The  emulsion  containing  99  per  cent  of 
kerosene  was  found  to  resemble  a  solid,  and  had  the  consistency 
of  blanc  mange. 

*  These  are  also  frequently  referred  to  as  the  continuous  and  discontinuous 
phases,  respectively. 

f  Jour.  Phys.  Chem.,  29,  407  (1916). 
J  Jour.  Chem.  Soc.,  91,  2002  (1907): 


COLLOIDS  255 

Characteristics  of  Gels.  Gels  are  generally  obtained  by  cooling 
or  evaporating  emulsoid  sols  and,  since  the  latter  are  known  to  be 
two-phase  liquid  systems,  it  is  natural  to  infer  that  gels  may  also 
be  two-phase  systems.  According  to  this  conception,  the  only 
difference  between  an  emulsoid  sol  and  a  gel  is,  that  in  the  latter 
the  concentration  of  at  least  one  of  the  phases  is  greatly  increased, 
and  thereby  imparts  greater  viscosity  and  rigidity  to  the  system. 
It  has  been  suggested,  that  the  more  concentrated  of  the  two  phases 
forms  the  walls  of  an  assemblage  of  cells  within  which  the  more 
dilute  phase  is  enclosed.  The  view  that  gels  possess  a  distinct 
cellular  structure  is  fully  confirmed  by  microscopic  examination. 

The  extreme  sensitiveness  of  gels  towards  changes  in  tempera- 
ture, and  towards  the  presence  of  extraneous  substances,  renders 
their  investigation  exceedingly  difficult.  Notwithstanding  the 
experimental  difficulties  involved  in  the  study  of  gels,  sufficient 
knowledge  of  their  properties  has  been  gained  to  make  a  brief 
account  of  these  necessary  in  any  treatment  of  the  subject  of 
colloids. 

Physical  Properties  of  Gels.  The  process  of  gel  formation  from  a 
dry,  gelatinous  colloid  and  water,  invariably  involves  contraction. 
This  statement  should  not  be  confused  with  the  fact,  that  a  gel, 
on  immersion  in  water,  undergoes  appreciable  increase  in  volume. 

Gels  have  been  shown  by  Barus*  to  be  considerably  more  com- 
pressible than  solids.  The  compressibility  increases  as  the  tem- 
perature is  raised  until,  when  the  gel  is  transformed  into  the  sol, 
the  compressibility  becomes  equal  to  that  of  pure  water.  The 
temperature  of  some  gels,  such  as  rubber  and  gelatine,  is  lowered 
by  compression  and  raised  by  tension. 

The.  thermal  expansion  of  gels  is  nearly  identical  with  that  of 
the  more  fluid  component  of  the  gel. 

The  rate  at  which  pure  substances  diffuse  in  gels  differs  only 
slightly  from  the  rate  of  diffusion  in  pure  water,  provided  the  con- 
centration of  the  gel  is  not  too  great.  The  slight  resistance  offered 
by  gels  to  diffusion  of  dissolved  substances,  may  be  regarded  as 
further  evidence  in  favor  of  their  cellular  structure. 

The  modulus  of  elasticity  of  a  gel,  cast  in  a  cylindrical  mold,  is 
given  by  the  formula 


*  Am.  Jour.  Sci.,  6,  285  (1898). 


256  THEORETICAL  CHEMISTRY 

where  P  is  the  tension  which  produces  the  increase  in  length  Al,  in 
a  cylinder  whose  length  is  I,  and  whose  radius  is  r.  It  has  been 
found  that  the  modulus  of  elasticity  in  gelatine  gels  increases  as 
the  square  of  the  concentration  of  the  gel.  The  time  of  recovery, 
after  releasing  the  tension,  increases  as  the  concentration  of  the 
gel  increases. 

The  shearing  modulus  for  a  gel  is  given  by  the  formula, 


where  /*  denotes  the  ratio  of  the  relative  contraction  of  the  diam- 
eter to  the  relative  change  in  length.  The  viscosity  of  a  gel  may 
be  calculated  from  the  shearing  modulus  by  means  of  the  equation 

1  =  E*T,  (11) 

where  r  denotes  the  time  of  recovery.  Since  both  Es  and  r  in- 
crease with  the  concentration  of  the  gel,  it  is  apparent  that  the  vis- 
cosity of  the  gel  must  also  increase  with  the  concentration. 

As  is  well  known,  when  glass  is  subjected  to  pressure  or  is  un- 
equally strained,  it  exhibits  the  phenomenon  of  double  refraction. 
Since  glass  bears  some  resemblance  to  gels  in  being  a  highly  vis- 
cous, supercooled  liquid,  it  might  reasonably  be  inferred  that  gels 
should  also  show  double  refraction.  Experiments  with  collodion 
and  gelatine  have  shown  that  these  substances,  when  subjected  to 
pressure,  behave  similarly  to  glass. 

Hydration  and  Dehydration  of  Gels.  The  complementary 
processes  of  hydration  and  dehydration  of  gels  are  extremely  inter- 
esting. Following  Freundlich,*  we  will  consider  the  subject  very 
briefly  under  the  two  following  heads:  (a)  Non-elastic  Gels,  and 
(b)  Elastic  Gels. 

(a)  Non-elastic  Gels.  When  freshly  prepared  aluminium,  or  fer- 
ric hydroxide  gels  were  placed  in  a  desiccator,  it  was  found  by 
van  Bemmelen  f  that  the  .rate  at  which  these  substances  lost 
water  was  continuous.  Furthermore,  on  removing  the  dried  gels 
from  the  desiccator  it  was  found  that  the  process  of  recovery  of 
moisture  was  also  continuous.  It  will  be  shown  in  a  subsequent 
chapter,  that  a  definite  hydrate,  in  the  presence  of  its  products 
of  dissociation,  possesses  a  constant  vapor  pressure,  so  long 

*  Kapillarchemie,  p.  486. 

t  Zeit.  anorg.  Chemie,  5,  466  (1894);  13,  233  (1897);  18,  14,  98  (1898); 
30,  265  (1902). 


COLLOIDS  257 

as  any  of  that  particular  hydrate  is  present.  The  fact  that  the 
vapor  pressure  of  the  gels  investigated  by  van  Bemmelen  did  not 
remain  constant,  but  decreased  continuously  as  the  water  was 
removed,  proved  conclusively  that  no  chemical  compounds  were 
formed  in  the  process  of  gel  hydration. 

Another  gel  which  has  been  made  the  subject  of  much  careful 
investigation  is  silicic  acid.  The  dehydration  curve  of  silicic 
acid  is  continuous,  with  the  exception  of  a  short  portion,  where  an 
appreciable  amount  of  water  is  lost  without  much  of  any  change  in 
the  vapor  pressure.  This  portion  of  the  curve  corresponds  to  a 
marked 'change  in  the  appearance  of  the  gel.  The  gel,  which  had 
hitherto  been  clear  and  transparent,  became  opalescent  soon  after 
the  vapor  pressure  had  attained  a  temporarily  constant  value. 
The  opalescence  gradually  permeated  the  entire  mass,  until  the 
gel  acquired  a  yellow  color  by  transmitted  light,  and  a  bluish 
color  by  reflected  light.  These  colors  suggest  a  marked  increase 
in  the  degree  of  dispersity  of  the  gel,  an  inference,  the  correctness 
of  which,  subsequent  investigation  has  fully  confirmed. 

The  curve  of  hydration,  while  resembling  the  curve  of  dehydra- 
tion in  many  respects,  departs  quite  widely  from  it  in  others. 
The  change  in  dispersity,  also  was  indicated  by  the  appearance, 
in  reverse  order,  of  the  color  and  opalescent  phenomena  men- 
tioned above. 

(6)  Elastic  Gels.  The  absence  in  elastic  gels  of  a  horizontal 
portion  in  the  dehydration  curves,  and  the  fact  that  although 
elastic  gels  may  have  become  saturated  with  water  vapor,  they 
still  retain  the  power  of  taking  up  large  amounts  of  liquid  water, 
when  immersed  in  that  medium,  constitute  the  chief  differences 
between  elastic  and  non-elastic  gels. 

The  amount  of  water  which  can  be  taken  up  by  an  elastic  gel  is 
exceedingly  large.  Thus,  on  exposing  a  plate  of  gelatine  weighing 
0.904  gram,  for  eight  days,  in  an  atmosphere  saturated  with  water 
vapor,  Schroeder  *  found  that  it  had  taken  up  0.37  gram  of  water. 
On  exposing  for  a  longer  period  of  time*  under  the  same  conditions, 
he  found  no  further  gain  in  weight,  and  on  removing  the  gelatine 
plate  from  the  moist  atmosphere  and  placing  it  in  a  desiccator,  the 
plate  slowly  gave  up  the  absorbed  moisture  and  regained  its  origi- 
nal weight.  On  the  other  hand,  when  the  plate,  after  having 
absorbed  the  maximum  weight  of  moisture  from  the  air,  was  im- 
*  Zeit.  phys.  Chem.,  45,  75  (1903). 


258 


THEORETICAL  CHEMISTRY 


mersed  in  water,  it  was  found  to  increase  in  weight  very  rapidly. 
Thus,  on  immersing  the  above  plate,  which  weighed  1.274  grams 
when  saturated  in  moist  air,  and  allowing  it  to  remain  in  water 
for  one  hour,  it  was  found  to  have  taken  up  5.63  grams  of  water. 
After  an  immersion  of  twenty-four  hours,  the  plate  was  found  to 
have  taken  up  the  maximum  weight  of  water  it  was  capable  of 
absorbing  at  that  temperature.  On  removing  the  plate,  it 
parted  with  the  absorbed  water  very  readily,  even  in  moist  air, 
the  greater  part  of  the  absorbed  water  being  so  loosely  held, 
that  the  vapor  pressure  of  the  gel  remained  the  same  as  that  of 
pure  water  at  the  same  temperature. 

It  is  impossible  to  measure  directly  the  pressure  produced  by 
gels  when  they  take  up  water,  but  some  idea  of  the  magnitude  of 
these  pressures  may  be  obtained  by  coating  a  glass  plate  with  gela- 
tine, which  has  imbibed  the  maximum  amount  of  water,  and  ob- 
serving the  degree  to  which  the  glass  plate  is  bent  by  the  drying 
gelatine  film.  Frequently  the  elastic  limit  of  the  glass  is  exceeded, 
and  the  plate  breaks  under  the  stress  produced  by  the  dehydration 
of  the  gel. 

Velocity  of  Imbibition.  The  rate  at  which  gels  imbibe  water 
has  been  studied  by  Hofmeister.*  The  velocity  of  imbibition  of 
water,  by  thin  plates  of  gelatine  and  agar-agar,  was  measured  by 
removing  the  immersed  plates  at  definite  intervals,  and  deter- 
mining their  increase  in  weight.  Owing  to  the  time  consumed 
in  making  the  weighings,  the  time  intervals  are  affected  by  an 
appreciable  error.  The  following  table  gives  the  data  of  a  single 
experiment  with  gelatine. 

VELOCITY  OF  IMBIBITION  IN  GELATINE 
(Thickness  of  plate  0.5  mm.) 


Time  (min.)- 

Water  Imbibed 

(grams). 

it 

5 
10 
15 
20 
25 

00 

3.08 
3.88 
4.26 
4.58 
4.67 
4.96 

0.090 
0.084 
0.084 
0.064 
0.075 

Arch.  exp.  Pathol.  u.  Pharmakol,  27,  395  (1890). 


COLLOIDS 


259 


If  the  weight  of  water  imbibed  in  t  minutes  is  wtj  and  if  w^  is  the 
maximum  weight  of  water  which  a  gel  can  take  up  under  the  con- 
ditions, then  the  velocity  of  imbibition  should  be  given  by  the 
equation, 

din 

(12) 


Tt  =  k  (w~  ~ 


which  on  integration  becomes, 

»-i 


-  Wt) 


(13) 


The  figures  in  the  third  column  of  the  preceding  table  were  cal- 
culated by  means  of  this  equation  and,  although  the  values  are  not 
strictly  constant,  the  variation  is  no  greater  than  might  be  ex- 
pected where  the  experimental  error  is  so  large. 

Heat  of  Imbibition.  The  process  of  imbibition  is  accom- 
panied by  an  evolution  of  heat.  Quantitative  measurements 
of  the  heat  evolved  when  gels  take  up  water,  have  been  made 
by  Wiedemann  and  Liideking,*  and  also  by  Rodewald.f  The 
following  table  gives  Rodewald's  data  on  the  heat  of  imbibition 
of  starch. 

HEAT  OF  IMBIBITION  OF  STARCH 

(Weight  of  dry  starch  100  grams.) 


Per  Cent,  Water. 

Heat  in  Calories  per 
Gram  of  Starch. 

0.23 

28.11 

2.39 

22.60 

6.27 

15.17 

11.65 

8.43 

15.68 

5.21 

19.52 

2.91 

It  will  be  observed,  that  the  greatest  development  of  heat  ac- 
companies the  initial  stages  of  imbibition,  where  very  small 
amounts  of  water  are  taken  up.  This  is  what  we  might  expect, 
when  we  remember,  that  it  is  the  last  remaining  portion  of  water 
which  is  most  difficult  "to  remove  from  a  gel,  and  that  it  is  only 
through  the  application  of  heat  that  its  complete  removal  can  be 
effected. 

*  Wied.  Ann.,  25,  145  (1885). 

t  Zeit.  phys.  Chem.,  24,  206  (1897). 


260  THEORETICAL  CHEMISTRY 

Imbibition  in  Solutions.  When  a  gel  is  immersed  in  a  saline 
solution,  the  salt  distributes  itself  between  the  solvent  and  the 
gel.  The  rate  of  imbibition  is  found  to  vary  greatly  according 
to  the  salt  which  is  present  in  the  solution.  Thus,  the  velocity 
of  imbibition  has  been  found  to  be  accelerated  by  the  presence 
of  the  chlorides  of  ammonium,  sodium  and  potassium,  and  also 
by  the  nitrate  and  bromide  of  sodium;  on  the  other  hand,  the 
presence  of  the  nitrate,  sulphate  and  tartrate  of  sodium  retard 
imbibition. 

The  effect  of  acids  and  bases  on  imbibition  appears  to  be  similar 
to  the  influence  of  salts. 

Adsorption.  The  tendency  exhibited  by  all  solids,  to  con- 
dense upon  their  surfaces  a  layer  of  any  gas  or  liquid  with 
which  they  may  be  in  contact,  is  termed  adsorption.  The 
amount  of  adsorption  is  conditioned  primarily  by  the  extent 
of  surface  exposed,  by  the  pressure  and  by  the  temperature. 
Under  the  same  conditions  of  temperature  and  pressure,  the 
amount  of  adsorption  varies  with  the  nature  of  the  adsorbent 
and  that  of  the  adsorbed  substance;  in  other  words,  adsorp- 
tion is  selective  or  specific. 

At  the  surface  of  a  solid  surrounded  by  a  gas  or  vapor,  the 
phenomenon  is  generally  known  as  gaseous  adsorption,  since  any 
difference  which  may  occur  in  the  concentration  of  the  solid  phase 
is  much  too  small  to  be  detected.  At  the  boundary  between 
liquid  and  gaseous  phases,  the  concentration  of  each  phase  un- 
doubtedly undergoes  alteration.  In  the  case  of  the  boundary  be- 
tween solid  and  liquid  phases,  the  only  apparent  inequality  in 
concentration  occurs  on  the  liquid  side  of  the  boundary,  notwith- 
standing the  fact  that  the  adsorbed  substance  is  quite  commonly 
regarded  as  being  bound  to  the  surface  of  the  solid  phase.  The 
cause  of  this  erroneous  conception  is,  that  the  extremely  thin  layer 
of  liquid  in  which  the  alteration  in  concentration  actually  occurs, 
is  the  layer  which  wets  the  surface  of  the  solid,  and  hence,  is  the 
layer  which  adheres  to  the  solid  when  it  is  removed  from  the 
liquid. 

The  retention  of  gases  by  charcoal  is  a  typical  example  of  gas- 
eous adsorption,  while  the  removal  of  coloring  matter  by  char- 
coal, in  the  purification  of  various  organic  substances,  may  be  cited 
as  an  example  of  adsorption  of  a  liquid  by  a  solid. 

If  the  adsorbed  substance  increases  in  concentration  in  the 


COLLOIDS 


261 


vicinity  of  the  boundary,  the  adsorption  is  said  to  be  positive;  if  it 
decreases,  the  adsorption  is  said  to  be  negative. 

When  any  substance  which  presents  a  large  extent  of  surface, 
such  as  charcoal  or  filter  paper,  is  brought  in  contact  with  a  gas 
or  a  solution,  the  process  of  adsorption  is  initiated,  and  will  con- 
tinue until  a  definite  amount  of  substance  has  been  adsorbed, 
or  until  adsorption-equilibrium  has  been  attained. 

Adsorption  of  Gases.  In  gases,  adsorption-equilibrium  is 
attained  with  remarkable  rapidity.  Thus,  if  a  gas  is  admitted 
into  a  vessel  containing  some  freshly  prepared  cocoanut  charcoal, 
the  pressure  will  fall  immediately  to  a  value  which  corresponds  to 
the  removal  of  the  entire  adsorbed  volume  of  gas. 

The  concentration  of  adsorbed  gas  on  the  surface  of  a  solid, 
when  equilibrium  is  attained,  has  been  shown  to  be  approximately 
1  X  10~7  gram  per  square 
centimeter.  This  value  is  of 
the  same  order  of  magnitude  as 
the  strength  of  the  limiting 
capillary  layer  of  a  liquid,  or, 
in  other  words,  the  adsorbed 
layer  of  gas  will  be  approx- 
imately one  molecule  deep. 

The  amount  of  gas  adsorbed  | 
by  a  solid  increases  with  the 
pressure  and  diminishes  with 
increasing  temperature.  If 
the  amount  of  adsorbed  gas 
be  plotted  against  the  cor- 
responding values  of  the  pres- 
sure, a  smooth  curve,  similar 
to  that  shown  in  Fig.  75  is 

obtained.     This    curve,    commonly    known    as    the    adsorption 
isotherm,  can  be  expressed  by  an  equation  of  the  form, 


Adsorption 
Fig.  75 


©" - *>• 


(14) 


where  x  is  the  amount  of  gas  adsorbed,  m  the  amount  of  the  ad- 
sorbing solid  and  p,  the  pressure,  while  k  and  n  are  experimentally 
determined  constants  which  vary  with  the  temperature.  This 
equation  enables  us  to  calculate  the  amount  of  adsorption  with  a 


262 


THEORETICAL  CHEMISTRY 


fair  amount  of  accuracy,  as  will  be  seen  in  the  accompanying  table 
giving  the  experimental  data  compiled  by  Geddes*  on  the  adsorp- 
tion of  carbon  dioxide  by  charcoal. 

ADSORPTION  OF  CARBON   DIOXIDE  BY  CHARCOAL 
(Temp.  31°;    n  =  1.77;    k  =  0.0602;    x  =  cc.     CO2  per  cc.  charcoal) 


p  (mm.  Hg.) 

x  (obs.) 

x  (calc.) 

p  (mm.  Hg.) 

x  (obs.) 

x  (calc.) 

41.5 

1.7 

1.7 

453 

6.5 

6.4 

120 

2.9 

3.1 

534 

7.1 

7.1 

194 

4.0 

4.0 

602 

7.6 

7.6 

276 

4.7 

4.9 

678 

8.3 

8.2 

340 

5.4 

5.5 

698 

8.6 

8.3 

405 

5.9 

6.0 

703 

8.9 

8.4 

Adsorption  in  Solutions.  The  adsorption  phenomena,  occur- 
ring at  the  surface  of  contact  of  a  solid  with  a  solution,  are  similar 
to  the  phenomena  which  have  just  been  discussed.  Because  of  the 
frequency  of  its  occurrence  in  many  of  the  more  common  opera- 
tions of  both  laboratory  and  factory,  the  subject  of  adsorption  in 
solutions  deserves  fuller  treatment. 

The  general  characteristics  of  adsorption  in  solutions  may  be 
briefly  summarized  as  follows :  — 

(1)  Adsorption  in  solutions  is  generally  positive,  i.e.,  on  shaking 
a  solution  with  a  finely-divided  adsorbent,  the  volume  concentra- 
tion of  the  solution  will  diminish. 

(2)  The  amount  of  positive  adsorption  may  be  sufficient  to  re- 
move almost  all  of  the  solute  from  a  solution,  especially  if  the  solu- 
tion is  dilute.     On  the  other  hand,  negative  adsorption  is  always 
very  small,  and  frequently  is  immeasurable. 

(3)  Adsorption  is  directly  proportional  to  the  so-called  "specific 
surface,"  the  latter  term  being  defined  as  the  ratio  of  the  total  sur- 
face of  the  adsorbent  to  its  volume. 

(4)  On  shaking  a  definite  weight  of  an  adsorbent  with  a  given 
volume  of  solution  of  known  concentration,  a  definite  equilibrium 
will  be  established.     If  the  solution  is  then  diluted  with  a  known 
amount  of  solvent,  the  adsorption  will  decrease  until  it  acquires 
the  same  value  which  it  would  have  attained,  had  the  same  weight 
of  adsorbent  been  introduced  directly  into  the  more  dilute  sola- 

*  Drude's  Ann.,  29,  797  (1909), 


COLLOIDS  263 

tion.  For  example,  if  1  gram  of  charcoal  is  agitated  with  100  cc. 
of  a  0.0688  molar  solution  of  acetic  acid  for  20  hours,  adsorption  is 
found  to  reduce  the  original  concentration  of  the  acid  to  0.0678 
molar.  In  a  second  experiment,  if  1  gram  of  charcoal  is  shaken  for 
the  same  period  of  time  with  50  cc.  of  2  X  0.0688  =  0.1376  molar 
acetic  acid,  and  then,  after  adding  50  cc.  of  water,  the  shaking  is 
continued  for  an  additional  period  of  3  hours,  the  final  concentra- 
tion of  the  acid  will  be  found  to  be  the  same  as  in  the  first  experi- 
ment. 

(5)  It  is  impossible  to  determine  the  specific  surface  of  an  ad- 
sorbent directly,  owing  to  its  porosity.     However,  according  to 
(3),  adsorption  is  directly  proportional  to  the  specific  surface,  and 
therefore,  the  weights  of  different  adsorbents  which  produce  the 
same  amount  of  adsorption  may  be  assumed  to  possess  equal  spe- 
cific surfaces. 

(6)  Adsorption  in  solution  is  largely  dependent  upon  the  surface 
tension  of  the  solvent.     In  solutions  of  the  same  substance  in 
different  solvents,  the  greatest  adsorption  occurs  in  that  solution 
whose  solvent  possessesTEe  Highest  surface  tension. 

(7)  The  order  of  efficiency  of  adsorption  is  not  only  independent 
of  the  nature  of  the  solvent,  but  also  of  the  nature  of  the  adsorbed 
substance. 

The  Adsorption  Isotherm.  The  empirical  equation  of  Freund- 
lich  for  gaseous  adsorption  has  been  found  to  apply  equally  well  to 
adsorption  equilibria  in  solutions.  The  equation  may  be  written 
as  follows: 

X~N"=fcC)  (15) 


where  x  is  the  weight  of  substance  adsorbed  by  a  weight  m,  of  ad- 
sorbent, from  a  solution  whose  volume-concentration  at  equilib- 
rium is  c,  and  where,  as  before,  k  and  n  are  constants.  The  con- 
stant n,  varies  in  different  cases  from  n  =  2  to  n  =  10;  within 
these  limits  the  value  of  n  is  independent  of  the  temperature,  and 
also  of  the  natures  of  the  adsorbed  substances  and  the  adsorbent. 
Although  the  value  of  the  constant  k,  varies  over  a  wide  range,  the 
ratio  of  its  values  for  two  adsorbents  in  different  solutions  is  prac- 
tically constant. 

The  following  table  contains  the  data  given  by  Freundlich  * 
on  the  adsorption  of  acetic  acid  by  charcoal. 
*  Kapillarchemie,  p.  147. 


264 


THEORETICAL  CHEMISTRY 


ADSORPTION   OF   AQUEOUS  ACETIC  ACID  BY  CHARCOAL 
t  =  25°;    k  =  2.606;    1/n  =  0.425 


Concentration 
(mola  per  liter). 

x/m  (obs.). 

x/m  (calc.). 

0.0181 

0.467 

0.474 

0.0309 

0.624 

0.596 

0.0616       - 

0.801 

0.798 

0.1259 

1.11 

1.08 

0.2677 

1.55 

1.49 

0.4711 

2.04 

1.89 

0.8817 

2.48 

2.47 

2.785 

3.76 

4.01 

The  validity  of  the  adsorption  isotherm  is  best  tested  graphic- 
ally, by  plotting  the  logarithms  of  the  experimentally  deter- 
mined values  of  x/m  against  the  logarithms  of  the  corresponding 
concentrations.  If  the  equation  holds,  a  straight  line  should  be 
obtained.  The  curves  shown  in  Fig.  76  represent  x/m  as  a  func- 
tion of  c,  and  log  x/m  as  a  function  of  log  c:  it  will  be  observed 
that  the  logarithmic  plot  is  practically  rectilinear. 

log  C 


*]£ 


C 
Fig.  76 

Surface  Energy  of  Colloids.  In  almost  all  colloidal  solutions 
there  exists  a  difference  of  potential  between  the  particles  of  the 
colloid  and  the  surrounding  medium.  The  importance  of  this 
factor  in  interpreting  the  behavior  of  colloids  has  already  been 
emphasized.  Another  factor  of  equal  importance  in  connection 
with  colloidal  phenomena,  is  that  which  depends  upon  the  enor- 


COLLOIDS  265 

mous  surface  of  contact  between  the  colloid  and  the  surrounding 
medium.  There  is  an  abundance  of  evidence  showing  that  a 
colloidal  solution  is  non-homogeneous,  or  in  other  words,  that  it 
is  essentially  a  suspension  of  finely-divided  particles  in  a  fluid 
medium.  An  immense  increase  in  superficial  area  results  from  the 
division  and  subdivision  of  matter.  To  bring  about  this  comminu- 
tion requires  a  large  expenditure  of  energy.  In  a  colloidal  solu- 
tion this  energy  is  stored  up  in  the  colloidal  particles  in  the  form 
of  surface  energy,  which  may  be  defined,  as  the  product  of  surface 
area  and  surface  tension. 

For  example,  suppose  1  cc.  of  a  substance  to  be  reduced  to 
cubical  particles  measuring  0.1  M  on  each  edge,  and  let  the  particles 
be  suspended  in  water  at  17°  C.  The  total  energy  involved  can 
be  calculated  as  follows :  —  The  volume  of  a  single  particle  is 
0.1  ju3  or  1  X  10~15  cc.,  hence  the  total  number  of  particles  is 
1  X  1015.  The  surface  of  a  single  particle  is  6  X  (0.1  /*)2,  or  6  X 
10~10  sq.  cm.,  and  the  total  surface  is  6  X  105  sq.  cm.  The  sur- 
face tension  of  water  at  17°  C.  is  71  dynes;  hence  the  total  sur- 
face energy  is  71  X  6  X  105  =  4.26  X  107  ergs.  This  enormous 
figure  shows  that  where  the  surface  of  the  disperse  phase  is  highly 
developed,  as  it  is  in  colloidal  solutions,  the  surface  energy  be- 
comes a  very  important  factor  in  determining  the  behavior  of  the 
system.  This  is  especially  the  case  when  the  degree  of  aggrega- 
tion of  the  colloidal  particles  is  changed,  since  a  relatively  small 
change  in  the  amount  of  aggregation  may  involve  a  great  change 
in  the  surface  exposed,  and  a  corresponding  change  in  the  surface 
energy.  A  very  close  connection  exists  between  the  electrical 
and  surface  factors  in  a  colloidal  solution. 

Surface  Concentration.  It  has  been  pointed  out  in  an  earlier 
chapter,  that  as  the  result  of  unbalanced  molecular  attraction,  the 
surface  of  a  liquid  behaves  like  a  tightly  stretched  membrane.  In 
consequence  of  this  contractile  force,  or  surface  tension,  the  pres- 
sure at  the  surface  of  a  liquid  is  greater  than  the  pressure  within 
the  liquid.  Furthermore,  when  a  dilute  solution  is  unequally 
heated,  the  solute  distributes  itself  in  accordance  with  the  gas 
laws,  the  solution  becoming  more  concentrated  in  the  cooler  por- 
tion. Just  as  the  homogeneity  of  a  dilute  solution  has  been  shown 
to  be  disturbed  by  inequality  of  temperature,  so  also  inequality 
of  pressure  may  be  assumed  to  cause  differences  in  concentration 
in  the  solution.  Although  direct  experimental  verification  is 


266  THEORETICAL  CHEMISTRY 

difficult,  there  is  abundant  evidence  for  the  view  that  the  concen- 
tration at  the  surface  of  solution  differs  from  the  volume-concen- 
tration of  the  solution,  in  consequence  of  the  greater  pressure  in  the 
surface  layer. 

The  mathematical  relation  between  surface  concentration  and 
surface  tension  was  first  deduced  by  J.  Willard  Gibbs  *  in  1876. 
The  following  simplified  derivation  of  this  important  equation  is 
due  to  Ostwald.  Let  s  be  the  surface  of  a  solution  whose  surface 
tension  is  y,  and  let  it  be  assumed  that  the  surface  contains  1  mol 
of  the  solute.  If  a  very  small  portion  of  the  solute  enters  the  sur- 
face layer  from  the  solution,  thereby  causing  a  diminution  dy,  in 
the  surface  tension,  the  corresponding  change  in  energy  will  be 
s  dy.  But  this  amount  of  energy  which  is  set  free,  must  be 
equivalent  to  the  osmotic  work  absorbed  in  effecting  the  removal 
of  the  same  weight  of  solute  from  the  solution.  Let  v  be  the 
volume  of  solution  containing  unit  weight  of  solute,  and  let  dp 
be  the  difference  in  the  osmotic  pressures  of  the  solution  before 
and  after  its  removal:  the  osmotic  work  will  be  —  v  dp.  Since 
the  gain  in  surface  energy  and  the  osmotic  work  are  equal,  we 

have 

s  dy  =  —  v  dp. 

The  solutions  being  dilute,  we  may  assume  that  the  gas  laws  hold, 
and  since  v  =  RT/p,  we  may  write, 

RT  , 
s  dy  =  --  dp, 

dy  RT 

or  -r-  =  --- 

dp  sp 

Since  pressure  is  directly  proportional  to  concentration,  the  pre- 
ceding equation  becomes 

dy=        RT 

dc  sc 

But  s  has  already  been  defined  as  the  surface  which  contains  1  mol 
of  solute  in  excess;  therefore  it  follows,  that  the  excess  of  solute 
in  unit  surface  is  1/s.  Writing  u  =  1/s,  we  have, 


which  is  the  equation  of  Gibbs. 

*  Trans.  Conn.  Acad.,  Vol.  Ill,  439  (1876). 


COLLOIDS  267 

From  this  equation  it  is  evident,  that  if  the  surface  tension,  7, 
increases  with  the  concentration,  then  u  is  negative,  and  the  sur- 
face concentration  is  less  than  the  concentration  of  the  bulk  of  the 
solution.  This  is  clearly  negative  adsorption.  On  the  other 
hand,  if  7  decreases  as  the  concentration  increases,  u  is  positive, 
and  the  surface  concentration  is  greater  than  the  concentration  of 
the  bulk  of  the  solution,  or  the  adsorption  is  positive.  Finally,  if 
the  surface  tension  is  independent  of  the  concentration,  then  the 
concentration  of  the  solute,  in  both  the  surface  layer  and  the 
bulk  of  the  solution,  will  be  the  same. 

The  Theory  of  von  Weimarn.  As  a  result  of  a  careful  study  of 
the  experimental  conditions  controlling  the  formation  and  exist- 
ence of  the  various  types  of  colloids,  von  Weimarn  *  has  formu- 
lated an  interesting  theory  as  to  the  process  of  formation  of  crystal- 
line precipitates.  In  order  to  understand  the  theory  of  von 
Weimarn,  let  us  consider  the  production  of  a  precipitate  in  a 
simple  metathetical  chemical  reaction,  taking  place  in  aqueous 
solution.  Let  us  assume  that  equivalent  concentrations  of  two 
substances,  which  are  non-associated  in  their  respective  solutions, 
react  to  form  two  other  non-associated  substances,  one  of  which 
is  precipitated,  while  the  other  remains  in  solution.  Let  L  be 
the  solubility,  in  mols  per  liter,  of  the  precipitated  substance, 
and  let  P  be  the  number  of  mols  of  the  precipitate  which  must 
separate  from  the  supersaturated  solution  in  order  that  its  con- 
centration may  be  reduced  to  the  value,  L  mols  per  liter.  Ob- 
viously, the  ratio,  P/L,  is  a  measure  of  the  degree  of  supersatura- 
tion  of  the  solution.  If  77  is  the  viscosity  of  the  solution,  then, 
according  to  von  Weimarn,  the  character  and  degree  of  dispersion 
of  the  precipitate  is  given  by  the  expression, 

*  =  £  fl,  (17) 

where  5  is  a  quantity,  known  as  the  dispersion  coefficient.  When 
a  series  of  precipitations  are  carried  out  in  such  a  manner  that 
the  value  of  6  is  the  same  in  each  case,  the  precipitates  are  said 
to  be  formed  under  "  corresponding  conditions."  According  to 
von  Weimarn,  the  law  governing  precipitation  under  correspond- 
ing conditions  may  be  expressed  as  follows :  —  The  general  char- 
acter and  the  degree  of  dispersion  of  crystalline  precipitates  are 
*  Griindziige  der  Dispersoidchemie. 


268 


THEORETICAL  CHEMISTRY 


always  the  same,  provided  precipitation  takes  place  under  corre- 
sponding conditions. 

The  significance  of  this  important  principle  and  the  applica- 
tion of  equation  (17)  to  a  typical  precipitation,  may  be  illustrated 
by  tabulating  the  results  obtained  by  von  Weimarn,  on  precipitat- 
ing barium  sulphate  from  mixtures  of  aqueous  solutions  of  barium 
thiocyanate  and  manganous  sulphate,  over  a  wide  range  of  con- 
centrations. 

VARIATION  IN  NATURE  OF  BaSO4  WITH  CONDITIONS 
OF   PRECIPITATION 


Equiv. 
Cone.,  of 
Reagents 

p 

gm.    per 
100  cc. 

d 

Nature  of   Precipitate 

0.00005 

0 

0 

No  ppt.  in  a  year  —  micro-crystals  to  be 

to 

expected  in  a  few  years. 

0.00014 

0.0006 

3 

0.00014 

0.0006 

3 

Slow  pp'tion  at  5  =  8.     Suspensoid  stage 

to 

at  6  =  25  (momentary).     Complete  sep- 

0.0017 

0.0096 

48 

aration  in  months  to  hours. 

0.0017 

0.0096 

48 

Precipitation  in  few  seconds  at  5    =    48. 

to 
0.75 

4.38 

21900 

Beyond  this,  instantaneous  precipitation. 
Crystal  skeletons  and  needles.     At  5  = 

21900,  crystals  barely  visible. 

0.75 

4.38 

21900 

Immediate   formation   of   amorphous  pre- 

3 

17.51 

87500 

cipitates. 

3 

17.51 

87500 

to 

Cellular,  clear  jelly. 

7 

40.9 

204500 

It  is  evident,  therefore,  that  the  degree  of  dispersion  of  the 
precipitate  is  greatest  when  low  and  high  concentrations  of  the 
reagents  are  employed,  and  that  the  particles  of  the  precipitate 
attain  their  maximum  size  in  the  region  of  medium  concentration. 

Preparation  of  Colloidal  Solutions.  Since  1861,  when  Graham 
published  his  first  paper  on  colloids,  numerous  investigators  have 
devised  methods  for  the  preparation  of  colloidal  solutions. 
Within  recent  years  our  knowledge  of  this  class  of  solutions  has 
been  greatly  increased,  many  crystalloidal  substances  having  been 
obtained  in  the  colloidal  condition.  As  a  result  of  these  investi- 
gations, we  no  longer  speak  of  crystalloidal  and  colloidal  matter, 


COLLOIDS  269 

but  use  the  terms,  crystalloid  and  colloid,  to  distinguish  two  dif- 
ferent states.  In  fact,  it  is  now  recognized  that  it  is  simply  a 
matter  of  overcoming  certain  experimental  difficulties,  before  it 
will  be  possible  to  obtain  all  forms  of  matter  in  the  colloidal 
state.  The  scope  of  this  book  forbids  a  detailed  account  of  the 
various  methods  which  have  been  devised  for  the  preparation  of 
colloidal  solutions.*  We  must  content  ourselves,  therefore,  with 
a  general  classification  of  these  methods  into  two  groups  as  fol- 
lows :  — 

(1)  Crystallization  Methods,  and  (2)  Solution  Methods. 
These  two  divisions  are  sufficiently  comprehensive  to  include  all 
of  the  known  methods  for  the  preparation  of  colloidal  solutions, 
with  the  possible  exception  of  the  electrical  methods  which  may 
be  considered  as  forming  a  separate  group. 

Crystallization  Methods.  The  crystallization  methods  include 
the  following  subdivisions :  — 

(1)  Methods  involving  cooling  of  a  liquid  or  solution. 
Example:  —  On   cooling   an   alcoholic   solution   of  sulphur  in 

liquid  air,  a  transparent,  highly  dispersed,  solid  sol  is  obtained. 

(2)  Methods  involving  change  of  medium. 

Example:  —  On  gradually  adding  a  solution  of  mastic  in  alcohol 
to  a  large  volume  of  water,  the  mastic  is  precipitated  in  a  finely 
divided  condition,  and  a  colloidal  mastic  hydrosol  results. 

(3)  Reduction  methods. 

Example:  —  On  adding  a  cold,  dilute  solution  of  hydrazine 
hydrate  to  a  dilute,  neutral  solution  of  auric  chloride,  a  dark  blue 
gold  sol  is  obtained. 

In  addition  to  hydrazine,  numerous  other  reducing  agents  may 
be  employed,  such  as  phosphorus,  carbon  monoxide,  hydrogen, 
acetylene,  formaldehyde,  acrolein,  various  carbohydrates,  hy- 
droxylamine,  phenylhydrazine,  and  metallic  ions. 

(4)  Oxidation  methods. 

Example:  —  On  oxidizing  a  solution  of  hydrogen  sulphide  by 
means  of  air  or  sulphur  dioxide,  a  colloidal  solution  of  sulphur  is 
obtained. 

(5)  Hydrolysis  methods. 

Example:  —  When  a  solution  of  ferric  chloride  is  added  slowly  to 

*  See  "  Die  Methoden  zur  Herstellung  Kolloider  Losungen  anorganischer 
Stoffe,"  by  The  Svedberg,  Dresden,  1909. 


270  THEORETICAL   CHEMISTRY 

a  large  volume  of  boiling  water,  the  salt  undergoes  hydrolysis, 
and  on  cooling  the  dilute  solution,  a  reddish-brown  ferric  hydrox- 
ide sol  is  obtained. 

(6)  Methods  involving  metathesis. 

Example:  —  A  colloidal  solution  of  silver  may  be  prepared,  by 
adding  a  few  drops  of  a  dilute  solution  of  sodium  chloride  to  a  di- 
lute solution  of  silver  nitrate,  provided  the  resulting  solution  of 
sodium  nitrate  is  below  the  precipitating  concentration. 

Solution  Methods.  Under  this  heading  are  to  be  grouped  the 
so-called  "  peptization "  methods.  The  term,  peptization,  was 
introduced  by  Graham  to  express  the  transformation  of  a  gel  into 
a  sol.  To-day,  we  understand  a  peptizer  to  be  a  substance  which,  if 
sufficiently  concentrated,  is  capable  of  effecting  the  solution  of  a 
solid  which  is  insoluble  in  its  dispersion  medium.  A  typical 
example  of  peptization  is  afforded  by  silver  chloride,  which  forms  a 
sol  on  prolonged  digestion  with  a  solution  containing  either  Ag* 
or  Cl'.  It  is  apparent  that  the  rate  of  peptization  can  be  con- 
trolled by  dilution  of  the  peptizer,  and  that  when  the  sol  stage  is 
attained,  the  peptizer  may  be  readily  removed  by  dialysis. 

Numerous  reactions  are  known,  in  which  the  conversion  of  an 
insoluble  precipitate  into  a  sol  can  be  effected  only  through  the 
removal  of  the  excess  of  electrolyte  by  prolonged  washing,  or  dial- 
ysis. A  familiar  example  of  this  type  of  peptization  is  furnished 
by  the  tendency  of  many  precipitates  to  run  through  the  filter 
after  too  prolonged  washing  with  water. 

Electrical  Methods.  These  methods  of  preparing  colloidal  solu- 
tions depend  upon  the  dispersive  action  of  a  powerful  electric 
discharge  upon  compact  metals.  In  1897,  Bredig  *  discovered, 
while  studying  the  action  of  the  electric  current  on  different 
liquids,  that  if  an  arc  be  established  between  two  metallic  wires 
immersed  in  a  liquid,  minute  particles  of  metal  are  torn  off  from 
the  negative  terminal  and  remain  suspended  in  the  liquid  indefi- 
nitely, f  In  order  to  prepare  a  colloidal  solution  by  the  method  of 
electrical  dispersion,  Bredig  recommends  that  a  direct  current 
arc  be  established  between  wires  of  the  metal  of  which  a  colloidal 
solution  is  desired,  the  ends  of  the  wires  being  submerged  in  water 

*  Zeit.  Elektrochem.,  4,  514  (1897);  Zeit.  phys.  Chem.,  31,  258  (1899). 

t  Recent  investigations  indicate  that  the  metal  of  the  electrode  is  vaporized 
by  the  current,  and  then  undergoes  condensation,  forming  an  extremely  stable 
suspension. 


COLLOIDS 


271 


Ammeter 


77 


in  a  well-cooled  vessel,  as  shown  in  Fig.  77.  The  current  em- 
ployed ranges  in  strength  from  5  to  10  amperes,  and  the  voltage 
lies  between  30  and  110  volts.  A  rheostat  and  an  ammeter  are 
included  in  the  circuit. 

The  wires  are  brought  in  contact  for  an  instant  in  order  to  es- 
tablish the  arc,  after  which  they  are  separated  about  2  mm. 
Clouds  of  colloidal  metal  are 
projected  out  into  the  water 
from  the  negative  wire,  a 
portion  of  the  metal  torn  off 
being  distributed  through  the 
water  as  a  coarse  suspension. 
The  size  of  the  particles 
disrupted  from  the  negative 
terminal  is  dependent  upon 
the  strength  of  the  current, 
a  current  of  10  amperes  pro- 
ducing a  greater  proportion 
of  colloidal  metal  than  a  cur- 
rent of  5  amperes.  The  addition  of  a  trace  of  potassium  hydroxide 
to  the  water  has  been  shown  to  facilitate  the  process  of  dispersion. 

When  gold  wires  are  used, 
deep  red  colloidal  solutions 
are  obtained,  which,  after  stand- 
ing for  several  weeks,  acquire 
a  bluish- violet  color.  With  extra 
precautions,  the  red  colloidal 
gold  solutions  may  be  pre- 
served for  two  years.  These 
solutions  have  been  shown  by 
Bredig  to  contain  about  14 
mg.  of  gold  per  100  cc.  In 
this  manner,  Bredig  prepared 
colloidal  solutions  of  platinum, 
palladium,  iridium,  and  silver. 
The  method  of  Bredig  has  been 
improved  and  extended  by  Svedberg. 

A  diagram  of  Svedberg's  apparatus  is  shown  in  Fig.  78.  The 
secondary  terminals  of  an  induction  coil,  capable  of  giving  a  spark 
ranging  from  12  to  15  cm.  in  length,  are  connected  in  parallel  with 


Induction  Coil 


78 


272  THEORETICAL  CHEMISTRY 

the  electrodes  and  a  glass  plate-condenser,  the  latter  having  a  sur- 
face of  approximately  225  sq.  cm.  Minute  fragments  or  grains  of 
the  metal  of  which  a  sol  is  desired,  are  placed  on  the  bottom  of  the 
vessel  containing  the  dispersion  medium.  The  electrodes,  which 
need  not  necessarily  be  of  the  same  metal,  are  immersed  as  shown 
in  the  diagram,  and  during  the  process  of  electrical  dispersion,  the 
contents  of  the  vessel  are  gently  stirred  with  one  or  the  other  of  the 
electrodes.  With  this  apparatus,  Svedberg  has  succeeded  in  pre- 
paring colloidal  solutions  of  tin,  gold,  silver,  copper,  lead,  zinc, 
cadmium,  carbon,  silicon,  selenium,  and  tellurium.  He  has  also 
obtained  all  of  the  alkali  metals  in  the  colloidal  state,  ethyl  ether 
being  used  as  the  dispersion  medium.  An  interesting  observation 
made  by  Svedberg  in  the  course  of  his  experiments  is,  that  the  color 
of  a  metal  is  the  same  in  both  the  colloidal  and  gaseous  states.* 

REFERENCES 

An  Introduction  to  the  Physics  and  Chemistry  of  Colloids.     Hatschek. 

The  Chemistry  of  Colloids.     Taylor. 

Theoretical  and  Applied  Colloid  Chemistry,  Wolfgang  Ostwald.  (Trans- 
lated by  Fischer.) 

Handbook  of  Colloid  Chemistry.  Wolfgang  Ostwald.  (Translated  by 
Fischer.) 

Applied  Colloid  Chemistry.     Bancroft. 

Surface  Tension  and  Surface  Energy.     Willows  and  Hatschek. 

Colloids  and  the  Ultramicroscope.     Zsigmondy.     (Translated  by  Alexander.) 

Atoms.     Perrin.     (Translated  by  Hammick.)     Chapters  II  to  IV  incl. 

Elements  and  Electrons.     Ramsay.     Chapters  VI  and  VII. 

Physical  Properties  of  Colloids.     Burton. 

Formation  of  Colloids.     Svedberg. 

Soaps  and  Protein's.     M.  Fischer. 

Laboratory  Manual  of  Colloid  Chemistry.     Holmes. 

*  For  a  description  of  Svedberg' s  latest  apparatus  the  student  is  referred  to 
his  book  entitled  "Formation  of  Colloids." 


CHAPTER  XI 
THERMOCHEMISTRY 

General  Introduction.  A  chemical  reaction  is  almost  invari- 
ably accompanied  by  a  thermal  change.  In  the  majority  of 
cases  heat  is  evolved;  a  violent  reaction  developing  a  large  amount 
of  heat,  while  a  feeble  reaction  develops  a  comparatively  small 
amount.  Such  reactions  are  said  to  be  exothermic.  A  relatively 
small  number  of  chemical  reactions  are  known  which  take  place 
with  an  absorption  of  heat.  These  are  termed  endothermic  reac- 
tions. Instances  of  chemical  reactions  unaccompanied  by  any 
thermal  change  are  very  rare,  and  are  almost  wholly  confined  to 
the  reciprocal  transformations  of  optical  isomers.  These  facts, 
which  were  first  observed  by  Boyle  and  Lavoisier,  led  to  the  view, 
that  the  amount  of  heat  evolved  in  a  chemical  reaction  might  be 
taken  as  a  measure  of  the  chemical  affinity  of  the  reacting  sub- 
stances. However,  with  the  advance  of  our  theoretical  knowl- 
edge, it  is  now  known  that  this  is  not  true,  although  a  parallelism 
between  heat  evolution  and  chemical  affinity  frequently  exists. 

Thermochemistry  is  concerned  with  the  thermal  changes  which 
accompany  chemical  reactions. 

Thermal  Units.  Heat  is  a  form  of  energy,  and,  like  other  forms 
of  energy,  it  may  be  resolved  into  two  factors;  an  intensity  fac- 
tor, the  temperature,  and  a  capacity  factor,  which  may  be  meas- 
ured in  any  one  of  several  units.  Among  these  units,  those  de- 
fined below  are  the  most  frequently  employed. 

The  small  calorie  (cal.)  is  the  quantity  of  heat  required  to  raise 
the  temperature  of  1  gram  of  water  from  15°  C.  to  16°  C.  The 
temperature  interval  is  specified  because  the  specific  heat  of  water 
varies  with  the  temperature.  The  large  or  kilogram  calorie  (Cal.), 
is  the  quantity  of  heat  required  to  raise  the  temperature  of  1000 
grams  of  water  from  15°  C.  to  16°  C.  The  Ostwald  or  average 
calorie  (K),  is  the  quantity  of  heat  required  to  raise  the  temper- 
ature of  1  gram  of  water  from  the  melting-point  of  ice  to  the  boil- 
ing-point of  water,  under  a  pressure  of  760  mm.  of  mercury.  It 

273 


274  THEORETICAL  CHEMISTRY 

is  approximately  equal  to  100  cal.,  or  to  0.1  Cal.  The  joule  (j), 
a  unit  based  on  the  C.G.S.  system,  is  equal  to  107  ergs.  This 
being  inconveniently  small  is  generally  multiplied  by  1000,  giving 
the  kilojoule  (J),  which  is,  therefore,  equal  to  1010  ergs.  The  last 
two  units  are  open  to  the  objection  that  their  values  are  depend- 
ent upon  the  mechanical  equivalent  of  heat,  any  change  in  the 
accepted  value  of  which  would  involve  a  correction  of  the  unit  of 
heat.  The  different  capacity  factors  of  heat  energy  are  related 
as  follows :  — 

1  cal.  =  0.001  Cal.  =  0.01  K  (approx.)  =  4.183  j  =  0.004183  J. 

Thermochemical  Equations.  In  order  to  represent  the  changes 
in  energy  which  accompany  chemical  reactions,  an  additional 
meaning  has  been  assigned  to  the  chemical  symbols.  As  ordina- 
rily used,  these  symbols  represent  only  the  molecular  or  formula 
weights  of  the  reacting  substances.  In  a  thermochemieal  or 
energy  equation  the  symbols  represent,  not  only  the  weight  in 
grams  expressed  by  the  formula  weights  of  the  substances,  but 
also  the  amount  of  heat  energy  contained  in  the  formula  weight 
in  one  state  as  compared  with  the  energy  contained  in  a  standard 
state.  For  example,  the  energy  equation, 

C  +  2  O  =  CO2  +  94,300  cal., 

indicates  that  the  energy  contained  in  12  grams  of  carbon  and 
32  grams  of  oxygen,  exceeds  the  energy  contained  in  44  grams  of 
carbon  dioxide,  at  the  same  temperature,  by  94,300  calories.  In 
writing  energy  equations,  it  is  very  essential  that  we  have  some 
means  of  distinguishing  between  the  different  states  of  aggrega- 
tion of  the  reacting  substances,  since  the  energy  content  of  a 
substance  is  not  the  same  in  the  gaseous,  liquid,  and  solid  states. 
In  the  system  proposed  by  Ostwald,  ordinary  type  is  used  for 
liquids,  heavy  type  for  solids,  and  italics  for  gases.  Another  and 
more  convenient  system  has  been  proposed,  in  which  solids  are 
designated  by  enclosing  the  symbol,  or  formula,  within  square 
brackets;  liquids  by  the  simple,  unbracketed  symbol,  or  formula; 
and  gases  by  enclosing  the  symbol,  or  formula,  within  parentheses. 
The  above  equation  should,  therefore,  be  written  in  the  following 
manner: 

[C]  +  (2  0)  =  (CO*)  +  94,300  cal. 


THERMOCHEMISTRY 


275 


Thermochemical  Measurements.  In  order  to  measure  the 
number  of  calories  evolved,  or  absorbed,  when  substances  react, 
it  is  necessary  that  the  reaction  should  proceed  rapidly  to  comple- 
tion. This  condition  is  fulfilled  by  two  classes  of  processes.  In 
the  first  class,  we  may  mention  the  processes  of  solution,  hydration, 
and  neutralization;  and  in  the  second  class,  the  process  of  com- 
bustion. 

The  apparatus  used  for  measuring  the  capacity  factor  of  heat 
energy  is  a  calorimeter.  This  instrument  consists,  essentially,  of 
a  large  insulated  vessel  containing  a  weighed  amount  of  water,  and 
a  smaller  submerged  vessel,  or  reaction  chamber,  within  which 
the  reaction,  whose  thermal  effect  is  sought,  is  allowed  to  take 
place.  By  observing  the  rise  in  temperature  of  the  water  accom- 
panying the  reaction,  and  knowing  the  heat  capacities  of  the 
calorimeter  and  its  accessories,  it  is  possible  to  calculate  the 
amount  of  heat  evolved,  or  absorbed,  in  the  reaction.  This  instru- 
ment may  be  given  a  variety  of  forms,  depending  upon  the  par- 
ticular use  to  which  it  is  to  be  put. 

The  Combustion  Calorimeter.  The  combustion  of  many  sub- 
stances, such  as  organic  compounds,  proceeds  very  slowly  in  air 
under  ordinary  pressures.  Such  reactions  can 
be  accelerated,  if  they  are  caused  to  take 
place  in  an  atmosphere  of  compressed  oxygen. 
For  this  purpose  the  combustion  calorimeter 
was  devised  by  Berthelot.*  In  this  apparatus 
the  essential  feature  is  the  so-called  combustion 
bomb,  shown  in  Fig.  79.  This  consists  of  a 
strong  cylinder  of  a  special  acid  resisting  alloy, 
which  is  furnished  with  a  heavy  threaded  cover. 
The  substance  to  be  burned  is  placed  in  a 
platinum  capsule  fastened  to  appropriate  sup- 
ports, while  a  weighed  piece  of  fine  iron  wire, 
the  middle  portion  of  which  dips  into  the  sub- 


Fig.  79 


stance,  is  connected  with  electric  terminals  on  the  cover  of  the 
bomb.  The  cover  is  screwed  down  tight,  and  the  bomb  is  filled 
with  oxygen,  under  a  pressure  of  from  20  to  25  atmospheres. 
The  bomb  is  then  submerged  in  the  calorimeter,  as  shown  at 
B  in  Fig.  80.  The  mass  of  water  in  the  calorimeter  being  known, 


*  Ann.  Chim.  Phys.,  (5),  23,  160  (1881);   (6),  10,  433  (1887). 


276 


THEORETICAL  CHEMISTRY 


and   its  temperature  having  been   read,  an  electric   current  is 
passed  through  the  iron  wire  in  the  bomb,  causing  it  to  burn 


M 


i 


.- 

I  L — j i~, 1» 


Fig.  80 

and  thus  ignite  the  substance.  The  rise  in  temperature,  due 
to  the  combustion,  is  observed,  and  the  quantity  of  heat  evolved 
is  calculated.  Corrections  must  be  applied  for  the  heat  evolved 


THERMOCHEMISTRY  277 

from  the  combustion  of  the  iron,  and  for  the  heat  evolved 
from  the  oxidation  of  the  nitrogen  of  the  residual  air  in  the 
bomb. 

The  calorimeter  shown  in  Fig.  80,  is  of  the  so-called  "  adiabatic  " 
type  in  which  the  temperature  of  the  jacketing  water  is  con- 
stantly maintained  at  the  same  temperature  as  that  of  the  water 
in  the  calorimeter  itself,  thereby  eliminating  the  troublesome  and 
uncertain  correction  for  loss  of  heat  by  radiation.  In  the  instru- 
ment illustrated  in  Fig.  80,  water  is  circulated  by  means  of  the 
stirrer  R,  which  includes  a  small  turbine  for  directing  a  portion 
of  the  water  upward  in  G,  and  thence  through  the  cover  C".  In 
this  manner  the  jacketing  water  is  uniformly  distributed  around 
the  vessel,  F,  and  the  bomb,  B.  The  control  of  the  temperature  is 
accomplished  by  having  available  a  supply  of  both  hot  and  cold 
water,  the  latter  coming  directly  from  the  laboratory  tap,  while 
the  former  is  furnished  by  a  small  heater,  (not  shown  in  the 
diagram),  located  at  the  right  of  the  apparatus.  The  hot  and 
cold  inlets  are  so  placed  that  the  control  cocks  are  easily  operated 
by  one  hand.* 

Law  of  Lavoisier  and  Laplace.  In  1780,  Lavoisier  and  Lap- 
lace, f  as  a  result  of  their  thermochemical  investigations,  enun- 
ciated the  following  law :  —  The  quantity  of  heat  which  is  required 
to  decompose  a  chemical  compound  is  precisely  equal  to  that  which 
was  evolved  in  the  formation  of  the  compound  from  its  elements.  This 
first  law  of  thermochemistry  will  be  seen  to  be  a  direct  corollary  of 
the  law  of  the  conservation  of  energy,  first  clearly  stated  by  Mayer 
in  1842. 

Law  of  Constant  Heat  Summation.  A  generalization  of  funda- 
mental importance  to  the  science  of  thermochemistry  was  discov- 
ered, in  1840,  bvjless.j  He  pointed  out,  that  the  heat  evolved  in  a 
chemical  process  is  the  same  whether  it  takes  place  in  one,  or  in  sev- 
eral steps.  This  is  known  as  the  law  of  constant  heat  summation. 
The  truth  of  the  law  may  be  illustrated  by  the  equality  of  the  heat 
of  formation  of  ammonium  chloride  in  aqueous  solution,  when 
prepared  in  two  different  ways. 

*  For  further  data  on  adiabatic  calorimetry  see  the  following  papers: 
Richards  and  Burgess,  Jour.  Am.  Chem.  Soc.,  32,  431  (1910);  Benedict  and 
Higgins,  ibid.,  32,  461  (1910). 

f  Oeuvres  de  Lavoisier,  Vol.  II,  p.  283. 

J  Pogg.  Ann.,  50,  385  (1841)). 


278  THEORETICAL  CHEMISTRY 

Thus, 

(A) 


(NH3)  +  (HC1)  =  [NH4C1]  +  42,100  cal. 

[NH4C1]  +  aq.    =  NH4C1,  aq.        -     3,900  cal. 

38,200  cal. 
(B) 

(NH3)  +  aq.  =  NH3,  aq.  +    8,400  cal. 

(HC1)  +  aq.  =  HC1,  aq.  +  17,300  cal. 

NH3,  aq.  +  HC1,  aq.  =  NH4C1,  aq.  +  12,300  cal. 

38,000  cal. 

It  will  be  observed,  that  the  total  amount  of  heat  evolved  in 
the  formation  and  solution  of  ammonium  chloride  is  the  same, 
within  the  limits  of  experimental  error,  whether  gaseous  ammonia 
and  hydrochloric  acid  are  allowed  to  react,  and  the  resulting  prod- 
uct is  dissolved  in  water;  or,  whether  the  gases  are  each  dissolved 
separately,  and  then  allowed  to  react.  It  should  be  noted,  that 
when  a  substance  is  dissolved  in  so  large  a  volume  of  water,  that 
the  addition  or  removal  of  a  small  portion  of  solvent  produces 
no  thermal  effect,  it  is  customary  to  denote  it  by  the  symbol  aq. 
(Latin  aqua  =  water.)  Thus, 

NH4C1,  aq.  +  nH2O  =  NH4C1,  aq., 
NH4C1,  aq.  -  nH2O  =  NH4C1,  aq. 

By  means  of  the  law  of  constant  heat  summation,  it  is  possible 
to  find  indirectly  the  amount  of  heat  developed,  or  absorbed,  by 
any  reaction,  even  though  it  is  impossible  to  carry  it  out  experimen- 
tally. For  example,  it  is  impossible  to  measure  the  heat  evolved 
when  carbon  burns  to  carbon  monoxide.  But  the  heat  evolved 
when  carbon  monoxide  burns  to  carbon  dioxide,  and  also  the  heat 
evolved  when  carbon  burns  to  carbon  dioxide,  can  be  accurately 
determined.  The  energy  equations  are  as  follows :  — 

[C]  +  2  (0)  =  (C02)  +  94,300  cal.  (1) 

(CO)  +     (0)  =  (CO2)  +  67,700  cal.  (2) 

Treating  these  equations  algebraically,  and  subtracting  equation 
(2)  from  equation  (1),  we  have 

[C].+  (0)  =  (CO)  +  2tf}600cal., 


THERMOCHEMISTRY  279 

or,  the  heat  of  combustion  of  carbon  to  carbon  monoxide  is  26,600 
calories.  Again,  as  a  further  illustration  of  the  applicability  of 
the  law  of  Hess,  we  may  take  the  calculation  of  the  heat  of  forma- 
tion of  hydriodic  acid  from  its  elements,  making  use  of  the  follow- 
ing energy  equations :  — 

2  KI,  aq.  +  2  (Cl)  =  2  KC1,  aq.  -f  2  [I]  +  524  K  (3) 

2  HI,  aq.  +  2  KOH,  aq.  =  2  KI,  aq.  +  2  H20  +  274  K  (4) 

2  HC1,  aq.  +  2  KOH,  aq.  =  2  KC1,  aq.  +  2  H2O  +  274  K  (5) 

2  (HI)  +  aq.  =  2  HI,  aq.  +  384  K,  (6) 

2  (HC1)  +  aq.  =  2  HC1,  aq. .+  346  K,  (7) 

2  (H)  +  2  (Cl)  =  2  (HC1)  +  440  K,  (8) 

adding  equations  (3)  and  (4), 

2  (Cl)  +  2  HI,  aq.  +  2  KOH,  aq.  =  2  KCl,  aq.  +  2  [I]  +  2  H2O 

/798K.  (9) 

Subtracting  equation  (5)  from  equation  (9), 

2  (Cl)  +  2  HI,  aq.  -  2  HC1,  aq.  =  2  [I]  +  524  K, 
or 

2  (Cl)  +  2  HI,  aq.  =  2  [I]  +  2  HC1,  aq.  +  524  K,     (10) 

adding  equations  (6)  and  (10), 

2  (HI)  +  aq.  +  2  (Cl)  =  2  [I]  +  2  HC1,  aq.  +  908  K,     (11) 
subtracting  equation  (7)  from  equation  (11), 

2  (HI)  H-  2  (Cl)  -  2  (HC1)  =  2  [I]  +  562  K, 
or 

2  (HI)  +  2  (Cl)  =  2  [I]  +  2  (HC1)  +  562  K,  (12) 

subtracting  equation  (8)  from  equation  (12), 

2  (HI)  -  2  (H)  =  2  [I]  +  122  K,  (13) 

or 

2  (H)  +  2  [I]  =  2  (HI)  -  122  K.  (14) 

In  a  similar  mannMMMjactically  any  heat  of  formation  may  be 
calculated,  provided  the  proper  energy  equations  are  com- 
bined. 


280  THEORETICAL  CHEMISTRY 

Heat  of  Formation.  The  intrinsic  energy  of  the  substances 
entering  into  chemical  reaction  is  unknown,  the  amount  of  heat 
evolved,  or  absorbed,  in  the  process  being  simply  a  measure  of  the 
difference  between  the  energy  of  the  reacting  substances  and  the 
energy  of  the  products  of  the  reaction.  Thus,  in  the  equation, 

[C]  +  2  (0)  =  (CO2)  +  94,300  cal., 

the  difference  between  the  energy  of  a  mixture  of  12  grams  of 
carbon  and  32  grams  of  oxygen,  on  the  one  hand,  and  the  energy 
of  44  grams  of  carbon  dioxide,  on  the  other,  is  seen  to  be  94,300 
calories.  The  equation  is  clearly  incomplete,  since  we  have  no 
means  of  determining  the  intrinsic  energies  of  free  carbon  and 
oxygen.  Furthermore,  since  the  elements  are  not  mutually  con- 
vertible, we  have  no  means  of  determining  the  difference  in  energy 
between  them.  It  is  customary,  therefore,  in  view  of  this  lack 
of  knowledge,  to  put  the  intrinsic  energies  of  the  elements  equal 
to  zero.^ 

If  the  heats  of  formation  of  the  substances  present  in  a  reac- 
tion are  known,  it  is  much  simpler  to  substitute  these  in  the  en- 
ergy equation,  and  solve  for  the  unknown  term.  This  method 
avoids  the  laborious  process  of  elimination  from  a  large  number  of 
energy  equations,  as  in  the  preceding  examples.  If  all  of  the  sub- 
stances involved  in  a  reaction  are  considered  as  decomposed  into 
their  elements,  it  is  evident  that  the  final  result  of  the  reaction 
will  be  the  difference  in  the  sums  of  the  heats  of  formation  on  the 
two  sides  of  the  equation.  This  leads  to  the  following  rule:  — 
To  find  the  quantity  of  heat  evolved,  or  absorbed,  in  a  chemical  reac- 
tion, subtract  the  sum  of  the  heats  of  formation  of  the  substances 
initially  present,  from  the  sum  of  the  heats  of  formation  of  the  products 
of  the  reaction,  placing  the  heat  of  formation  of  all  elements  equal  to 
zero. 

The  energy  equation  for  the  formation  of  carbon  dioxide  from 
its  elements  may  then  be  written  as  follows :  — 

0  +  0  =  (CO2)  +  94,300  cal., 
or 

(CO2)  =  -  94,300  cal. 

Therefore,  in  writing  an  energy  equatior™  Hike  use  of  tlje  fol- 
lowing rule :  —  Replace  the  formulas  of  e JH  mpound  in  the  reac- 


THERMOCHEMISTRY  281 

\     4- 

lion  equation  by  the  negative  values  of  their  respective  heats  of 
formation,  and  solve  far  ike  unknown  term.  This  unknown  term 
may  be  either  the  heat  of  a  reaction,  or  the  heat  of  formation 
of  one  of  the  reacting  substances.  The  following  examples  will 
serve  to  illustrate  the  application  of  the  above  rules :  — 

•* 

(1)  Let  it  be  required  to  find  the  heat  of  the  following  reaction 

[MgCl2]  +  2  [Na]  =  2  [NaCl]  +  [Mg]  +  x, 

where  x  is  the  heat  of  the  reaction.  The  heat  of  formation  of 
MgCl2  is  151  Cal.,  and  that  of  NaCl  is  97.9  Cal.,  therefore, 

-151  -t  0  =  -  (2  X  97.9)  +  0  +  z, 
or 

x  =  44.8  Cal. 

(2)  The  heat  of  combustion  of  1  mol  of  methane  is  213.8  Cal., 
and  the  heats  of  formation  of  the  products,  carbon  dioxide  and 
water,  are  94.3  Cal.  and  68.3  Cal.,  respectively.     Let  it  be  required 
to  find  the  heat  of  formation  of  methane.     Representing  the  heat 
of  formation  of  methane  by  x,  we  have 

(CH4)  +  2  (02)  =  (C02)  +  2  (H20)  +  213.8  Cal., 
-  x  +  0  =  -  94.3  -  2  X  68.3  +  213.8, 

or 

x  =  17.1  Cal. 

(3)  The  heat  of  combustion  of  1  mol  of  carbon  disulphide  is 
265.1  Cal.,  the  thermochemical  equation  being, 

CS2  +  3  (Os)  =  (CO2)  +  2  (SOa)  +  265.1  Cal. 

The  heats  of  formation  of  carbon  dioxide  and  sulphur  dioxide  are 
94.3  Cal.  and  71  Cal.,  respectively.  The  heat  of  formation  of 
carbon  disulphide,  x,  may  then  be  calculated  as  follows: 

_  x  +  o  =  -  94.3  -  2  X  71  +  265.1, 

or  f 

.8  Cal. 


x  =  428.! 


Carbon  disulphide  is  thus  seen  to  be  an  endothermic  compound. 
The  heats  of  formation  of  a  number  of  metallic  and  non-metallic 
compounds  are  given  in  the  accompanying  table. 


282 


THEORETICAL  CHEMISTRY 


HEATS  OF  FORMATION  * 
(METALLIC  COMPOUNDS) 


Oxides 

Na,O...  .  97,100cal. 

K20 91,000 

CaO 145,000 

BaO 133,400 

MgO 143,600 

ZnO 85,440 

FeO 65,700 

Fe2O3 197,000 

Fe3O4 270,800 

PbO 50,300 

CuO 37,200 

Ag2O 5,900-7,000 

HgO 21,500 

Sulphides 

Na2S..  .  89,300  cal. 

K2S 103,500 

CaS 90.800 

CuS 10,000 

PbS 18,400 

Chlorides 

NaCl..  .  97, 700  cal. 

KC1 105,600 

BaCl2 196,900 

MgCl2 151,000 

ZnCl2 97,200 

FeCl2 82,050 

FeCl3 96,040 

CuCl2 51,630 

AgCl 29,900 

PbCl2 82,800 

Bromides 

NaBr 85,700  cal. 

KBr 95,300 

AgBr.... 22,700 

PbBr2..      64,450 


Iodides 

Nal..  .  61,900  cal. 

KI 80,130 

Cul 16,300 

Agl : 15,000 

PbI2 39,800 

Nitrates 

NaNO3..  ..111,000  cal. 

KNO3 119,000 

Ba(NO3)2 228,000 

AgNO3 28,740 

Pb(NO3)2 105,500 

Sulphates 

Na2SO4..  ..  .328,590  cal. 

K2SO4 344,300 

BaSO4 340,000 

MgSO4 302,000 

ZnSO4 229,600 

CuSO4 182,600 

Ag2SO4 167,300 

PbSO4 216,210 

(Non-metallic   Compounds) 

HC1..  .  22,000  cal. 

HBr 8,440 

HI -6,040 

H2O  (liq.) -68,360 

H2O  (gas) 58,000 

NH3 11,890 

SOa... 71,080 

H2S04 192,920 

N2O -17,740 

NO -21,570 

HNO3 41,600 

CO 29,000 

CO2 96,960 

CH4 21,750 

C2H2 -47,770 

C6H6 -12,510 


*  Data  compiled  from  Physikalish-chemische  Tabellen,  Landolt-Born- 
stein-Meyerhoffer. 

Heat  of  Solution.  The  thermal  change  accompanying  the 
solution  of  1  mol  of  a  substance,  in  so  large  a  volume  of  solvent 
that  subsequent  dilution  of  the  solution  causes  no  further  change  in 
temperature,  is  termed  the  heat  of  solution.  The  solutJQ_n_of  neutral 
salts  is  generally  an  endothermic  process.  This  fact  may  be 
readily  accounted  for,  on  the  hypothesis  that  considerable  heat 


THERMOCHEMISTRY 


283 


must  be  absorbed,  as  heat  of  fusion  and  heat  of  vaporization,  be- 
fore the  solid  salt  can  assume  a  condition  in  solution  which  closely 
resembles  that  of  a  gas.  The  heat  of  solution  of  hydrated  salts 
is  less  than  the  heat  of  solution  of  the  corresponding  anhydrous 
salts.  For  example,  the  heat  of  solution  of  1  mol  of  anhydrous 
calcium  nitrate  is  4000  calories,  while  the  heat  of  solution  of  1  mol 
of  the  tetrahydrate  is  —  7600  calories. 

The  difference  between  the  heats  of  solution  of  the  anhydrous 
and  hydrated  salts  is  termed  the  heat  of  hydration.  The  heats  of 
solution  in  water  of  some  of  the  more  common  compounds  are 
given  in  the  following  table,  the  values  being  expressed  in  small 
calories. 

HEATS  OF  SOLUTION 


Substance 

Mols 
H20 

Q 

Substance 

Mols 
H20 

Q 

HC1 

300 

+  17,315  cal. 

NaBr... 

200 

-       190  cal. 

HBr 

400 

+  19,940 

NaBr.2H2O.... 

300 

-    4,710 

HI 

500 

+  19  210 

Nal 

200 

+    1,220 

NH3 

200 

+    8  430 

Nal  2H2O   .  .  . 

300 

4,010 

H2SO4 

1600 

+  17  850 

NaOH 

200 

+    9,940 

HNO3 

300 

+    7,480 

NaNO3. 

200 

-    5,030 

NH4C1  
NH4NO3.... 
NaCl  

200 
200 
100 

-    3,880 
-    6,320 
-    1,180 

Na2SO4  
Na2SO4  10H2O 
KC1  

400 
400 
200 

+       460 
-  18,760 
-    4,440 

Ca(OH)2 

2500 

+    2,790 

KBr  

200 

-    5,080 

CaCl2  
CaCl2.6H2O 
ZnSO4.  .  .      . 

300 
400 
400 

+  17,410 
-    4,310 
+  18,430 

KI  
KOH  
KNO3  

200 
250 
200 

-    5,110 
+  13,290 
-    8,520 

ZnSO4  7H2O 

400 

-    4,260 

K2SO4  

400 

-    6,380 

CuCl2  

600 

+  11,080 

CuSO4  

400 

+  15,800 

CuGl2.2H2O 

400 

+    4,210 

CuSO4.5H2O... 

400 

-    2,750 

Heat  of  Dilution.  The  heat  of  dilution  of  a  solution  is 
the  quantity  of  heat  which  is  evolved,  or  absorbed,  per  mol  of  solute 
when  the  solution  is  greatly  diluted.  Beyond  a  certain  dilution, 
further  addition  of  solvent  produces  no  thermal  change.  While 
there  is  a  definite  heat  of  solution  for  a  particular  solute  in  a  par- 
ticular solvent,  the  heat  of  dilution  remains  indefinite,  since  the 
latter  is  dependent  upon  the  degree  of  dilution.  Those  gases 
which  obey  Henry's  law,  are  practically  the  only  substances  which 
have  no  appreciable  heats  of  solution,  or  dilution. 

The  following  tables  give  the  heats  of  dilution  of  several  acids 
and  bases. 


284 


THEORETICAL  CHEMISTRY 
HEATS  OF  DILUTION  OF  ACIDS 


Mols  of  Water  to 
1  mol  Acid 

HC1 

HNOa 

H2S04 

1 
2 
5 

8 

5,370  cal. 
11,360 
14,960 

3,340  cal. 
4,860 
6,600 
7,220 

6,380  cal. 
9,420 
13,110 

10 
20 
50 
100 
200 
300 

16,160 
16,760 
17,100 
17,200 

17  300 

7,270 
7,360 

7,210 
7,180 

14,950 
16,260 
16,700 
16,900 
17,100 

800 

17,600 

1600 

17,900 

HEATS  OF  DILUTION  OF  BASES 


Mols  of  Water 
Added 

To  KOH  +  3  H2O 

To  NaOH  +  3  H2O 

2 

1500  cal. 

2130  cal. 

4 

2100 

2890 

6 

2360 

3100 

17 

2700 

3080 

47 

2740 

3100 

97 

2750 

3000 

197 

2750 

2940 

It  is  apparent,  that  as  the  dilution,  of  the  above  solutions  is  in- 
creased, the  thermal  effect  pro- 
duced by  the  addition  of  more 
solvent  becomes  less  and  less,  and 
ultimately,  becomes  zero.  This 
is  shown  diagrammatically  in  Fig. 
81,  in  which  the  abscissas  repre- 
sent the  composition  of  the  solu- 
tions, and  the  ordinates  represent 
the  corresponding  heats  of  solu- 
tion. The  ordinates,  An\  and 
Ariz,  are  the  heats  of  solution 
of  one  mol  of  A,  in  n\  and  n^ 
mols  of  B,  respectively.  The 
difference  between  these  two  heats 


mols  of  B  to  1  mol  of  A 

Fig.  81 


of  solution,  is  represented  by  BC:  this  is  the  so-called  "  integral 


THERMOCHEMISTRY  285 

heat  of  dilution/'  corresponding  to  an  increase  in  the  number  of 
mols  of  B,  from  n\  to  n^..  If  one  mol  of  B  is  added  to  a  large 
volume  of  solution,  containing  one  mol  of  A  in  HI  mols  of  B,  the 
resulting  thermal  change  is  called  the  "differential  heat  of  dilu- 
tion," and  will  be  represented  by  the  tangent  to  the  curve  at  A. 

Reactions  at  Constant  Volume.  When  a  chemical  reaction 
takes  place  without  any  change  in  volume,  or  when  the  external 
work,  resulting  from  a  change  in  volume,  is  not  included  in  the 
heat  of  the  reaction,  the  process  is  said  to  take  place  at  constant 
volume.  That  is  to  say,  the  condition  of  constant  volume,  is  a 
condition  which  involves  no  external  work,  either  positive,  or 
negative.  Under  these  conditions,  the  total  energy  of  the  react- 
ing substances  is  equal  to  the  total  energy  of  the  products  of  the 
reaction,  plus  the  quantity  of  heat  developed  by  the  reaction, 

Reactions  at  Constant  Pressure.  When  a  chemical  reaction 
is  accompanied  by  a  change  in  volume,  the  system  suffers  a  loss 
of  heat  equivalent  to  the  work  done  against  the  atmosphere,  if 
the  volume  increases;  or,  the  system  gains  an  amount  of  heat 
equivalent  to  the  work  done  on  the  system  by  the  atmosphere, 
if  the  volume  decreases.  Under  these  conditions,  the  reaction  is 
said  to  take  place  at  constant  pressure.  The  difference  between 
constant  volume  and  constant  pressure  conditions  is,  that 
under  the  former,  the  heat  equivalent  of  the  work  corresponding 
to  any  change  in  volume  which  may  occur,  is  not  considered  as 
having  any  effect  upon  the  energy  of  the  system;  whereas,  under 
the  latter,  due  account  is  taken  of  the  change  in  energy  resulting 
from  change  in  volume.  Suppose  that  in  a  reaction,  1  mol  of 
gas  is  formed.  Under  standard  conditions  of  temperature  and 
pressure,  the  volume  of  the  system  will  be  increased  by  22.4  liters. 

The  formation  of  gas  involves  the  performance  of  work  against 
the  atmosphere,  this  work  being  done  at  the  expense  of  the  heat 
energy  of  the  system.  To  calculate  the  heat  equivalent  of  this 
work,  let  us  imagine  the  gas  enclosed  in  a  cylinder,  fitted  with 
a  piston  whose  area  is  1  square  centimeter.  The  normal  pressure 
of  the  atmosphere  on  the  piston  is  76  cm.  of  mercury,  or  1033.3 
grams  per  square  centimeter.  If  the  increase  in  the  volume  of 
the  gas  is  22.4  liters,  the  piston  will  be  raised  through  22,400  cm., 
and  the  work  done  will  be,  1033.3  X  22,400  gram-centimeters. 
The  heat  equivalent  of  this  change  in  volume  will  be,  (1033.3  X 
22,400)  -f-  42,600  =  542.3  calories,  or  0.5423  large  calories.  There- 


286  THEORETICAL  CHEMISTRY 

fore,  this  amount  of  heat  must  be  added  to  the  heat  of  the  reaction. 
It  should  be  observed  that  this  correction  is  independent  of  the 
actual  value  of  the  pressure  upon  the  system.  Thus,  if  the  pres- 
sure is  increased  n  times,  the  volume  of  the  gas  will  be  reduced  to 
l/n  of  its  former  value,  and  the  work  done  will  involve  moving 
the  piston  through  l/n  of  the  distance  against  an  ft -fold  pressure, 
which  is  plainly  equivalent  to  the  former  amount  of  work.  While 
the  correction  is  independent  of  the  pressure,  it  is  not  independent 
of  the  temperature.  The  familiar  equation,  pv  —  RT,  shows  us 
that  the  work  done  by  a  gas  is  directly  proportional  to  its  ab- 
solute temperature.  Thus,  if  a  gas  is  evolved  at  273°,  it  will 
occupy  double  the  volume  it  would  occupy  at  0°,  and  the  work 
done  at  273°  will  involve  moving  the  piston  through  twice  the 
distance  that  it  would  have  to  be  moved  at  0°.  Theoretically,  a 
gas  evolved  at  the  absolute  zero  would  occupy  no  volume,  and 
hence  no  work  would  be  done.  Introducing  the  correction  for 
temperature,  we  see  that 

XT=  1.986  Teal., 

must  be  added  to  the  heat  of  the  reaction,  where  T  is  the  absolute 
temperature  at  which  the  change  in  volume  occurs.  For  all 
ordinary  purposes,  it  is  sufficiently  accurate  to  take  2  T  calories 
as  the  correction.  Thus,  suppose  n  mols  of  gas  to  be  formed  in  a 
reaction  at  17°  C.:  the  amount  of  heat  absorbed  will  be 

n  X  2  (273  +  17)  =  580  n  cal. 

Under  constant  pressure  conditions,  the  symbols,  in  addition  to 
their  usual  significance,  represent  the  energy,  plus  or  minus  the 
term,  2  T  per  mol,  the  positive  sign  being  used  if  gas  is  absorbed, 
while  a  negative  sign  is  used  if  gas  is  set  free.  The  constant  vol- 
ume condition  is  a  condition  in  which  no  account  is  taken  of  the 
external  work,  even  if  a  change  in  volume  does  occur  during  the 
reaction,  while  the  constant  pressure  condition  .is  one  in  which 
the  external  work  is  taken  into  consideration.  It  is  apparent 
therefore,  that  the  relation  of  the  heat  energy  of  a  reaction  at 
constant  volume,  QV)  to  the  heat  energy  at  constant  pressure,  QP) 
can  be  represented  by  the  equation, 

Qv  =  Qv-2nT  cal.,  (15) 


THERMOCHEMISTRY  -        287 

where  n  denotes  the  number  of  mols  of  gas  formed  in  excess  of 
those  initially  present.  This  equation  is  of  great  importance  in 
connection  with  the  determination  of  heats  of  combustion  in  the 
bomb-calorimeter,  in  which  the  reactions  necessarily  take  place 
under  constant  volume  conditions.  Since  it  is  customary  to  state 
heats  of  reaction  under  constant  pressure  conditions,  the  foregoing 
equation  makes  it  possible  to  convert  heats  of  combustion,  deter- 
mined under  constant  volume  conditions,  into  heats  of  combustion 
under  constant  pressure  conditions.  For  example,  the  com- 
bustion of  naphthalene  takes  place  in  accordance  with  the  equa- 
tion, 

[CioHg]  +  12  (02)  =  10  (C02)  +  4  (H20)  +  1242.95  Cal. 

It  is  apparent,  that  the  reaction  is  accompanied  by  an  increase  in 
volume,  due  to  the  formation  of  2  mols  of  gas;  therefore  at  15°  C. 
the  correction  will  be, 

Q9  =  1242.95  -  [2  X  0.002  (273  +  15)], 
or,  Qp  =  1241.8  Cal. 

The  volume  occupied  by  solids  or  liquids  is  so  small  as  to 
be  negligible,  and  does  not  enter  into  these  calculations. 

Variation  of  Specific  Heat  with  Temperature.  In  a  previous 
chapter,  (p.  50)  it  has  been  shown  that  the  value  of  the  molec- 
ular specific  heat  of  a  monatomic  gas,  at  constant  pressure,  is 
5.0.  According  to  the  kinetic  theory  of  gases,  the  heat  capacity, 
at  constant  pressure,  of  any  monatomic  gas  not  only  should  be 
constant,  but  also  should  be  independent  of  the  temperature. 
This  theoretical  requirement  has  been  fully  confirmed  by  experi- 
ments carried  out  at  temperatures  ranging  from  —  80°  to  200°. 
The  kinetic  theory  also  indicates  that  the  molal  heat  capacity 
of  gases,  containing  more  than  one  molecule,  should  increase  with 
the  molecular  complexity  of  the  gas,  and  also  with  the  temper- 
ature. Among  the  earliest  investigators  in  this  field  was  Le  Chat- 
elier,*  who  showed  that  the  variation  of  the  molecular  specific 
heat  of  polyatomic  gases  with  temperature  could  be  expressed  by 
the  formula, 

CP  =  6.5  +  aT,  (16) 

in  which  the  value  of  the  constant,  a,  increases  with  the  complex- 
ity of  the  molecule.  It  was  subsequently  shown  by  Nernst,t 

*  Zeit.  phys.  Chem.,  i,  546  (1887). 
t  Gottinger  Nachrichten,  i,  1906, 


288  THEORETICAL  CHEMISTRY 

that  the  equation  of  Le  Chatelier  does  not  hold  at  high  tempera- 
tures, and  he  proposed  a  modified  expression  of  the  form, 

Cp  =  3.5  +  1.5n  +  a!T,  (17) 

where  n  denotes  the  number  of  atoms  in  the  molecule. 

After  testing  a  large  number  of  empirical  formulas  which  have 
been  proposed  from  time  to  time,  to  express  the  variation  of  the 
molal  heat  capacity  of  gases  with  temperature,  Lewis  and  Ran- 
dall* have  selected  the  following  as  the  most  trustworthy  :  — 

For  N2,  02,  CO,  NO,  HC1,  HBr,  and  HI, 

Cp  =  6.50  +  0.0010  T:  (18) 

For  H2, 

Cp  =  6.50  +  0.0009  T:  (19) 

For  C12,  Br2,  and  I2, 

Cp  =  6.5  +  0.004  T:  (20) 

For  H2O  and  H2S, 

Cp  =  8.81  -  0.0019  T  +  0.00000222  T2:  (21) 

For  C02  and  SO2, 

Cp  =  7.0  +  0.0071  T  -  0.00000186  T2:  (22) 

For  NH3, 

Cp  =  7.5  +  0.0042  T.  (23) 

The  use  of  these  equations  may  be  illustrated  by  calculating  the 
quantity  of  heat  required  to  raise  the  temperature  of  one  mol  of 
carbon  dioxide  from  50°  to  100°.  To  calculate  the  quantity  of 
heat  absorbed,  it  is  necessary  to  integrate  the  equation,  Q  =  CpdT, 
between  the  desired  temperature  limits,  making  use  of  equation 
(22)  to  determine  the  value  of  Cp,  as  follows  : 

Q  =    C      (7.0  -  0.0071  T  +  0.00000186  T2)  dT, 
Jm° 


=  7.0  (373  -  323)  -  (3732  -  3232) 

.      +  0.00000186  (3,33.3,33) 

o 

=  350  -  123.54  +  11.28  =  237.74  cal. 

It  will  be  observed  that  all  of  the  foregoing  formulas  are  de- 
rived for  constant  pressure  conditions.     They  may  readily  be 

*  Jour.  Am.  Chem.  Soc.,  34,  1128  (1912). 


THERMOCHEMISTRY 


289 


transformed,  however,  so  as  to  apply  equally  well  to  constant 
volume  conditions,  by  subtracting  2  calories  from  the  first  term  of 
each  equation.  Thus,  equation  (18)  may  be  written  as  follows: 

C,  =  4.50  +  0.0010  T,  (24) 

which  will  give  the  molal  heat  capacity  of  N2,  Oz,  CO,  NO,  HC1, 
HBr,  and  HI  at  constant  volume,  since  the  molal  heat  capacities 
of  gases  at  constant  volume  are  known  to  be  approximately  2 
calories  per  degree  less  than  the  corresponding  values  at  constant 
pressure. 

Similar  expressions  for  the  heat  capacity  of  solids  in  terms  of 
temperature  have  already  been  briefly  discussed  in  an  earlier  chap- 
ter, (p.  99). 

Up  to  the  present  time  no  satisfactory  formulas  have  been 
derived  to  express  the  relation  of  the  thermal  capacities  of  liquids 
to  temperature. 

Variation  of  Heat  of  Reaction  with  Temperature.  If  a  chem- 
ical reaction  be  allowed  to  take  place,  first  at  the  temperature 
ti,  and  then  at  the  temperature  ^,  the  amounts  of  heat  developed 
in  the  two  cases  will  be  found  to  be  quite  different.  Let  us  assume 
that  the  initial  state  of  the  system  is  represented  by  the  point 
A,  in  the  diagram,  Fig.  82,  and  that  the  final  state  is  represented 


B 


m 


Fig.  82 


by  the  point  B.  It  is  evident  that  we  can  pass  from  A  to  B  by 
two  independent  paths,  namely  AnB  and  AmB,  According  to 
the  first  law  of  thermodynamics,  the  net  absorption,  or  evolution, 
of  heat  along  AnB  must  be  the  same  as  that  along  AmB,  since 
the  system  starts  from  the  point  A  and  ends,  in  both  cases,  at  the 
same  point  B,  without  undergoing  any  change  in  volume. 

Let  Qi  and  $2  represent  the  quantities  of  heat  evolved  at  the 
temperatures,  ti  and  t2,  respectively.     Let  us  imagine  that  the 


290  THEORETICAL  CHEMISTRY 

reaction  takes  place  at  the  temperature  fa,  Qi  units  of  heat  being 
evolved;  and  then  let  the  products  of  the  reaction  be  heated  to 
the  temperature  k.  If  c'  represents  the  total  thermal  capacity 
of  the  products  of  the  reaction,  then  the  quantity  of  heat  neces- 
sary to  produce  this  rise  in  temperature  will  be  ce  (k  —  ti).  Now 
let  us  imagine  the  reacting  substances,  at  the  temperature  h,  to 
be  heated  to  the  temperature  tz,  and  then  allowed  to  react,  with  the 
evolution  of  Q2  units  of  heat.  The  heat  necessary  to  produce  this 
rise  in  temperature  in  the  reacting  substances  is,  c  (t2  —  ti),  where 
c  is  the  total  thermal  capacity  of  the  original  substances.  Hav- 
ing started  with  the  same  substances  at  the  same  initial  temper- 
ature, and  having  obtained  the  same  products  at  the  same  final 
temperature,  we  have,  according  to  the  first  law  of  thermodyna- 
mics, 

Qi  -  c'  (fe  -  *i)  =  Q2  -  c  (fe  -  *i), 
or 

Qt-Oi  _  ,  _  ,' 

~  " 


or,  where  the  change  in  temperature  is  very  small, 

f=c-C'.  (24) 

If  c'  is  greater  than  c  then  the  sign  of  dQ/dt  will  be  negative,  or,  in 
other  words,  an  increase  in  temperature  will  cause  a  decrease  in 
the  heat  of  reaction.  On  the  other  hand,  if  c  is  greater  than  c', 
dQ/dt  will  be  positive  and  the  heat  of  reaction  will  increase  with 
the  temperature. 

EXAMPLE.     The   reaction   between   hydrogen   and   oxygen   at 
18°  C.  is  represented  by  the  following  equation: 

2  (H8)  +  (02)  =  2  H2O  +  1367.1  K. 

Suppose  it  is  required  to  find  how  much  heat  will  be  evolved  when 
the  two  gases  react  at  110°  C.,  the  product  of  the  reaction  being 
maintained  at  this  -temperature,  and  the  pressure  remaining  con- 
stant. The  specific  heats  of  the  different  substances  involved, 
are  as  follows: 
Hydrogen  =  3^409;  Oxygen  =  0.2175;  Water  (betweenx  18° 

and  100°)  =  1;  (Water  between  100°  and  110°)  =  0.5. 
The  heat  of  vaporization  of  water  is  537  calories  per  gram. 


THERMOCHEMISTRY  29 1 

For  liquid  water  per  degree,  we  have,  ^ 

dQ/dt  =  (4  X  3.409  +  32  X  0.2175)  -  (36  X  1)  =  -  15.404  caL, 

and   for    (100°  -  18°)  =  82°,    we   have,    82  X  (  -  15.404)  =  - 
1263  cal.     The  heat  of  formation  of  liquid  water  at  100°  is,  there- 
fore, 

1367.1  -  12.-63  =  1354.47  K. 

When  the  liquid  water  is  vaporized  at  100°,  (36  X  537)  calories  of 
this  heat  is  absorbed,  or  the  formation  of  steam  at  100°  from  hy- 
drogen and  oxygen,  evolves 

1354.47  -  193.32  =  1161,15  K. 
For  steam  per  degree,  we  have, 

dQ/dt  =  (4  X  3.409  +  32  X  0.2175)  -  (36  X  0.5)  =  2.596  cal., 
and  for  the  interval  (110°  -  100°)  =  10°, 

10  X  2.596  =  25.96  cal. 
Or,  for  the  total  heat  evolved,  we  have 

1161.15  +  0.2596  =  1161.41  K. 

Heat  of  Combustion.  The  heat  evolved  during  the  complete 
oxidation  of  unit  mass  of  a  substance  is  termed  its  heat  oj 
combustion.  The  unit  of  mass  commonly  chosen  in  all  physico- 
chemical  calculations  is  the  mol.  An  enormous  amount  of  ex- 
perimental work  has  been  done  on  the  determination  of  the  heats 
of  combustion  of  a  large  number  of  chemical  compounds  by  Thorn- 
sen,*  Berthelbt,t  LangbeinJ  Richards  §  and  others.  The  accom- 
panying table  gives  the  heats  of  combustion  of  several  organic 
compounds  recently  determined  with  extreme  accuracy  by  Rich- 
ards and  Davis. 

*  Thermochemische  Untersuchungen,  4  Vols. 

t  Essai  de  Mecanique  Chimique,  Thermochimie,  Donees  et  Lois  Numeri- 
ques. 

t  Joiir.  prakt.  Chem.,  1885  to  1895. 

.->!§  Richards  and  Barry,  Jour.  Am.  Chem.  Soc.,  37,  993  (1915);  Richards 
and  Davis,  ibid.,  42,  159£  (1920). 


292  THEORETICAL  CHEMISTRY 

HEATS  OF  COMBUSTION  AT  CONSTANT  VOLUME 


Substance 

Heat  of  Combustion  per 
Mol 

Sucrose 

1  349  400  cal. 

Benzoic  acid  
Naphthalene   .  . 

771,550 
1  231  600 

Benzene  

781,850 

Toluene  

935,230 

Tertiary  butyl  benzene. 
Cyclohexanol  

1,399,800 
889,300 

Di-isoamyl 

1  612  600 

Methyl  alcohol 

170  610 

Ethyl  alcohol  

327,040 

Propyl  alcohol  
Butyl  alcohol  

485,800 
638,330 

Isobutyl  alcohol 

637  140 

It  has  been  observed  that  the  average  increase  in  the  molal 
heat  of  combustion,  in  any  ascending  homologous  series  of  hydro- 
carbons, is  153,500  calories  for  each  CH2  group  added.  It  has 
recently  been  pointed  out  by  Thornton,*  that  the  molal  heat  of 
combustion  of  any  saturated  hydrocarbon  is  approximately  52,700 
calories  for  each  atomic  weight  of  oxygen  required  to  burn  it. 
For  example,  the  complete  combustion  of  methane  may  be  repre- 
sented by  the  equation, 

(CH4)  +2(02)  =  (C02)  +2H20; 

since  four  atomic  weights  of  oxygen  are  required  to  burn  one  mol 
of  methane,  it  follows,  according  to  Thornton's  rule,  that  the 
heat  of  combustion  of  the  latter  must  be,  4  X  52,700  =  210,800 
calories,  a  value  which  is  in  exact  agreement  with  that  obtained 
by  direct  experiment. 

A  number  of  other  interesting  relations  between  heats  of  com- 
bustion of  compounds  and  their  differences  in  composition  have 
been  discovered,  but  these  cannot  be  taken  up  at  this  time.  It 
has  also  been  pointed  out,  that  the  heat  of  combustion  of  organic 
compounds  is  conditioned  not  only  by  their  composition,  but  also 
by  their  molecular  constitution,  f 

Some  exceedingly  interesting  and  important  results  have  been 

*  Phil.  Mag.,  33,  196  (1917). 

t  Redgrove,  Chem.  News,  116,  37  (1917);  Swietoslawski,  Jour.  Am. 
Chem.  Soc.,  42,  1312  (1920). 


THERMOCHEMISTRY  293 

obtained  with  the  different  allotropic  forms  of  the  elements.  For 
example,  when  equal  masses  of  the  three  common  allotropic  forms 
of  carbon  are  burned  successively  in  oxygen,  the  amounts  of  heat 
evolved  are  found  to  be  quite  different,  as  is  shown  by  the  fol- 
lowing energy  equations: 

[C]  diamond         +  2  (O)  =  (C02)  +  94.3    Cal. 

[C]  graphite         +  2  (O)  =  (CO2)  +  94.8    Cal. 

[C]  amorphous     +  2  (O)  =  (CO2)  +  97.65  Cal. 

It  is  apparent,  that  amorphous  carbon  contains  the  greatest 
amount  of  energy  of  any  one  of  the  three  allotropic  modifications, 
and  therefore,  if  this  form  of  carbon  were  to  undergo  transforma- 
tion into  the  diamond,  the  reaction  would  be  accompanied  by  the 
evolution  of  (97.65  -  94.3)  =  3.35  Cal.  In  like  manner,  the  allo- 
tropic forms  of  both  sulphur  and  phosphorus  have  different  heats 
of  combustion.  The  following  equations  express  the  differences 
in  intrinsic  energy  between  the  allotropic  forms  of  the  two 
elements : 

S  (monoclinic)  =  S  (rhombic)  +  2.3  Cal. 
P  (white)  =  P  (red)  +  3.71  Cal. 

When  the  same  substance  is  burned  in  oxygen,  and  then  in  ozone, 
it  is  found,  that  more  heat  is  evolved  in  ozone  than  in  oxygen. 
The  energy  equation  expressing  the  change  of  ozone  into  oxygen 
may  be  written  thus, 

(O,)  =  li  (O2)  +  36.2  Cal. 

All  of  the  above  facts  illustrate  the  general  principle,  that  larger 
amounts  of  intrinsic  energy  are  associated  with  unstable  than 
with  stable  forms. 

Thermoneutrality  of  Salt  Solutions.  In  addition  to  the  law 
of  constant  heat  summation,  Hess  discovered  two  other  important 
laws  of  thermochemistry,  viz.,  the  law  of  thermoneutrality  of 
salt  solutions,  and  the  law  governing  the  neutralization  of  acids 
by  bases.*  When  two  dilute  salt  solutions  are  mixed  there  is 
neither  evolution  nor  absorption  of  heat.  Thus  when  dilute  solu- 
tions of  sodium  nitrate  and  potassium  chloride  are  mixed,  there  is 
no  thermal  effect.  The  energy  equation  may  be  written  as  fol- 
lows: — 

NaN03,  aq.  +  KC1,  aq.  =  NaCl,  aq.  +  KN03,  aq.  +  0  Cal. 
*  Pogg.  Ann.,  50,  385  (1840). 


294  THEORETICAL  CHEMISTRY 

According  to  this  equation,  a  double  decomposition  has  taken 
place,  and  we  should  naturally  expect  an  evolution,  or  an  absorp- 
tion, of  heat.  While  Hess  could  not  account  for  the  absence  of  any 
thermal  effect,  he  recognized  the  fact  as  quite  general,  and  formu- 
lated the  law  of  the  thermoneutrality  of  salt  solutions,  as  fol- 
lows :  —  The  metathesis  of  neutral  salts  in  dilute  solutions  takes  place 
with  neither  evolution  nor  absorption  of  heat. 

The  explanation  of  the  phenomenon  of  thermoneutrality  was 
furnished  by  the  theory  of  electrolytic  dissociation.  When  the 
above  equation  is  written  in  the  ionic  form,  it  becomes 

Na'  +  NCV  +  K'  +  Cr  =  Na*  +  Cl'  +  K'  +  NOV. 

From  this  it  is  apparent,  that  the  same  ions  exist  on  both  sides  of 
the  equation,  and  in  reality  no  reaction  takes  place. 

There  are  numerous  exceptions  to  the  law  of  thermoneutrality, 
but  these  also  can  be  satisfactorily  accounted  for  by  means  of  the 
theory  of  electrolytic  dissociation.  All  of  those  salts,  the  behavior 
of  which  in  dilute  solution  is  contrary  to  the  law,  are  found  to  be 
only  partially  ionized,  and,  therefore,  when  their  solutions  are 
mixed,  a  chemical  reaction  actually  occurs.  The  exceptions  must 
be  considered  as  furnishing  additional  evidence  in  favor  of  the 
theory  of  electrolytic  dissociation. 

Heat  of  Neutralization.  Hess  also  discovered,*  that  when 
dilute  solutions  of  equivalent  quantities  of  strong  acids  and 
strong  bases  are  mixed,  practically  the  same  amount  of  heat  is 
evolved  in  each  case.  The  following  energy  equations  may  be 
considered  as  typical  examples  of  such  neutralizations: 

HC1,  aq.  +  NaOH,  aq.  =  NaCl,  aq.  +  H2O  +  13.68  CaL, 
HNO3,  aq.  +  NaOH,  aq.  =  NaNO3  aq.  +  H20  +  13.69  CaL, 

HC1,  aq.  +  KOH,  aq.  =  KC1,  aq.  +  H20  +  13.93  CaL, 
HN03,  aq.  +  KOH,  aq.  =  KN03,  aq.  +  H2O  +  13.87  CaL, 

HC1,  aq.  +  LiOH,  aq.  =  LiCl,  aq.  +  H20  +  13.70  CaL 

Here  again,  it  would  be  difficult  to  explain  the  phenomenon 
without  the  theory  of  electrolytic  dissociation.  By  means  of  this 
theory,  however,  the  explanation  is  both  simple  and  satisfactory. 
If  MOH  and  HA  represent  any  strong  base  and  any  strong  acid, 
respectively,  then  when  equivalent  amounts  of  these  are  dissolved 

*  Loc.  cit. 


THERMOCHEMISTRY  295 

in  water,  each  solution  being  largely  diluted  to  the  same  volume, 
the  reaction  may  be  written  thus: 

M'  +  OH'  +  H*  +  A'  =  M*  +  A'  +  H2O  +  13.8  Cal. 

Disregarding  the  ions  which  occur  on  both  sides  of  the  equality 
sign,  we  have 

OH'  +  H*  =  H2O  +  13.8  Cal. 


It  thus  appears,  that  trie'  neutralizationol  a  strong  acid  by  a  strong 
base  in  dilute  solution,  consists  solely  in  the  combination  of  hydro- 
gen and  hydroxyl  ions  to  form  undissociated  water.  The  heat 
of  this  ionic  reaction  is  13.8  large  calories.  The  heat  of  formatipn 
of  water  from  its  ions,  must  not  be  confused  with  the  heat  of 
formation  of  water  from  its  elements. 

When  weak  acids,  or  weak  bases  are  neutralized  by  strong  bases, 
or  strong  acids,  or  when  weak  acids  are  neutralized  by  weak  bases, 
the  heat  of  neutralization  may  differ  widely  from  13.8  Cal.  This 
is  shown  by  the  following  thermochemical  equations: 

H-COOH,  aq.  +  NaOH,  aq.  =  H-COONa,  aq.  +  H2O 

+  13.40  Cal., 
CHC12-COOH,  aq.  +  NaOH,  aq.  =  CHCl2-COONa,  aq.  +  H2O 

+  14.83  Cal., 
H-COOH,  aq.  +  NH4OH,  aq.  =  H-COONH4,  aq.  +  H2O 

+  11.90  Cal., 
HCN,  aq.  +  NaOH,  aq.  =  NaCN,  aq.  +  H2O  +  2.90  Cal. 

As  will  be  seen,  the  heat  of  neutralization  may  be  either  greater, 
or  less,  than  13.8  Cal.  The  exceptions  to  the  rule,  that  the  heat 
of  neutralization  of  an  acid  by  a  base  is  constant,  are  readily  ex- 
plained by  the  theory  of  electrolytic  dissociation.  Suppose  a  weak 
acid  to  be  neutralized  by  a  strong  base.  According  to  the  dissoci- 
ation theory,  the  acid  is  only  slightly  dissociated,  and  therefore, 
yields  a  comparatively  small  number  of  hydrogen  ions  to  the 
solution.  The  base  on  the  other  hand,  is  completely  dissociated 
into  hydroxyl  and  metallic  ions.  Therefore,  as  many  hydroxyl 
ions  disappear  as  there  are  free  hydrogen  ions  with  which  they 
can  combine,  to  form  water.  When  the  equilibrium  between  the 
acid  and  the  products  of  its  dissociation  has  been  thus  disturbed, 
it  undergoes  further  dissociation,  and  the  resulting  hydrogen  ions 
immediately  combine  with  the  free  hydroxyl  ions  of  the  base. 


296 


THEORETICAL  CHEMISTRY 


This  process  continues,  until  all  of  the  hydroxyl  ions  of  the  base 
have  been  neutralized.  It  is  evident  that  the  total  thermal  effect, 
in  this  case,  must  be  equal  to  the  algebraic  sum  of  the  heat  of  dis- 
sociation of  the  weak  acid,  which  may  be  positive  or  negative, 
and  the  heat  of  formation  of  water  from  its  ions.  A  similar  ex- 
planation holds  for  the  neutralization  of  a  weak  base  by  a  strong 
acid,  or  for  the  neutralization  of  a  weak  acid  by  a  weak  base. 

The  approximate  value  of  the  heat  of  dissociation  of  a  weak 
acid,  or  a  weak  base  may  be  estimated,  provided  its  heat  neutral- 
ization by  a  strong  base,  or  strong  acid  is  known.  For  example, 
in  the  equation  given  above, 

HCN,  aq.  +  NaOH,  aq.  =  NaCN,  aq.  +  H2O  +  2.90  Cal., 

the  difference  between  2.9  and  13.8,  or  —  10.9.  Cal.,  represents 
approximately,  the  heat  of  dissociation  of  hydrocyanic  acid. 
Since  the  acid  is  initially  slightly  dissociated  in  dilute  solution, 
it  is  apparent,  that  in  order  to  obtain  the  true  heat  of  dissociation, 
we  must  add  to  —  10.9  Cal.,  the  thermal  value  of  the  dissociation 
of  that  portion  of  the  acid  which  has  already  become  ionized. 

It  has  been  shown  by  Wormann,*  that  the  average  value  of 
13.8  kilogram  calories  per  mol  of  water  formed,  does  not  remain 
constant  at  temperatures  much  above,  or  below,  room  temperature. 
The  following  table  contains  some  of  Wormann's  data. 

HEATS  OF  NEUTRALIZATION  AT   DIFFERENT 
TEMPERATURES 


Temperature 

Concentration  of 
Acid  and  Base 

KOH  +  HC1 

NaOH  +  HC1 

0° 

0.25 

14,707  cal. 

14,580  cal. 

0 

0.10 

14,709 

14,604 

6 

0.25 

14,473 

14,352 

6 

0.1875 

14,463 

14,359 

6 

0.125 

14,448 

14,331 

18 

0.25 

13,937 

13,714 

18 

0.125 

14,957 

13,693 

18 

0.05 

13,887 

13,631 

32 

0.25 

13,155 

12,974 

32 

0.125 

13,171 

12,922 

32 

0.05 

13,160 

12,980 

*  Ann.  der  Physik,  18,  775  (1905). 


THERMOCHEMISTRY 


297 


Heat  of  lonization.  Since  13.8  Cal.  is  the  heat  of  formation 
of  water  from  its  ions,  this  must  also  be  the  thermal  equivalent  of 
the  energy  required  to  dissociate  one  mol  of  water  into  its  ions.  It 
must  be  remembered  that  the  dissociated  molecule  of  water  must 
be  mixed  with  a  very  large  volume  of  undissociated  water,  in  order 
that  the  dissociation  may  be  permanent.  Reference  to  the  table 
of  heats  of  formation  (p.  282),  will  show  that  68.4  Cal.  are  required 
to  form  one  mol  of  water  from  its  elements.  Hence,  it  follows  that 
68.4  —  13.8  =  54.6  Cal.,  is  the  heat  of  formation  of  one  equiva-  ' 
lent  of  hydrogen  and  hydroxyl  ions.  It  has  been  shown  that  an 
extremely  small  amount  of  energy  is  necessary  to  ionize  hydrogen 
when  it  is  dissolved  in  water.  It  is  evident,  therefore,  that  54.6 
Cal.  is  a  close  approximation  to  the  heat  of  formation  of  one  equiv- 
alent of  hydroxyl  ions. 

On  the  assumption  that  the  heat  of  ionization  of  gaseous  hydro- 
gen in  solution  is  zero,  the  values  of  the  other  ionic  heats  of  forma- 
tion may  be  computed.  For  example,  the  heat  of  formation  of 
KOH,  aq.  is  116.5  Cal.  The  heat  of  formation  of  potassium 
ions  must  be,  116.5  —  54.6  =  61.9  Cal.  In  like  manner,  the 
heat  of  formation  of  KC1,  aq.  is  101.2  Cal.;  hence  the  heat 
of  formation  of  chlorine  ions  must  be,  101.2  —  61.9  =  39.3  Cal. 
The  accompanying  table  of  ionic  heats  of  formation  has  been  calcu- 
lated in  this  manner. 


HEAT  OF  FORMATION  OF  IONS 


Ion. 

Heat  of 
Formation. 

Ion. 

Heat  of 
Formation. 

Hydrogen  
Potassium 

0.0 
61.9 
57.5 
62.9 
*  32.8 
109.0 
109.0 
121.0 
50.2 
22.2 
-9.3 
17.0 
16.0 
35.1 
18.4 

i  Copper  (ic)  

-15.8 
-16.0 
-19.8 
-25.3 
0.5 
3.3 
39.3 
28.2 
13.1 
214  .4 
151.3 
27.0 
49.0 
161.1 
54.6 

Copper  (ous)  .  . 

Sodium 

jj  *t~    \       / 
Mercury  (ous)  
Silver  
Lead  
Tin  (ous)  

Lithium 

Ammonium  . 

IVlagnesium 

Calcium 

Chlorine  

Aluminium  
M^anganese 

Bromine                

Iodine  

Iron  (ous) 

Sulphate  
Sulphite  

Iron  (ic) 

Cobalt 

Nitrous  

Nickel 

Nitric  

Zinc  

Carbonate  
Hydroxyl  

Cadmium  

298  THEORETICAL  CHEMISTRY 

The  Principle  of  Maximum  Work.  A  fundamental  principle 
of  the  science  of  mechanics  is,  that  a  system  is  in  stable  equilib- 
rium when  its  potential  energy  is  a  minimum.  In  1879,  Ber- 
thelot  *  suggested  that  a  similar  principle  applies  to  chemical 
systems. 

In  terms  of  the  kinetic  theory,  the  temperature  of  a  substance 
is  to  be  regarded  as  a  measure  of  the  kinetic  energy  of  its  mole- 
cules. The  development  of  heat  by  a  chemical  reaction  would, 
therefore,  be  taken  as  an  indication  of  a  decrease  in  the  potential 
energy  of  the  system.  Berthelot's  theorem,  known  as  the  prin- 
ciple of  maximum  work,  may  be  stated  as  follows:  —  "  Every  chem- 
ical reaction  which  proceeds  to  completion  without  the  intervention  of 
energy  from  an  external  source,  tends  to  produce  that  substance,  or 
system  of  substances,  which  evolves  the  maximum  amount  of  heat" 
The  table  of  heats  of  formation  (p.  282)  illustrates  the  general 
truth  of  this  principle,  but  as  will  be  seen,  the  theorem  precludes 
the  possibility  of  spontaneous  endothermic  reactions.  Thus,  for 
example,  the  formation  of  acetylene  from  its  elements  at  the  tem- 
perature of  the  electric  arc  is  a  well-known  endothermic  reaction, 
but  according  to  the  principle  of  maximum  work,  it  would  not 
take  place  spontaneously.  Another  serious  objection  to  Berthe- 
lot's principle  is,  that  according  to  it,  all  chemical  reactions  should 
proceed  to  completion,  the  reaction  taking  place  in  such  a  way  as 
to  evolve  the  greatest  amount  of  heat.  As  is  well  known,  many 
reactions,  and  theoretically  all  reactions,  are  never  complete,  but 
proceed  until  a  condition  of  equilibrium  is  reached.  The  principle 
of  maximum  work,  therefore,  denies  the  existence  of  equilibria  in 
chemical  reactions.  Many  attempts  have  been  made  to  "  explain 
away  "  these  defects,  but  none  of  them  have  been  successful.  In 
referring  to  the  generalization,  Le  Chatelier  terms  it  "  a  very  in- 
teresting approximation  toward  a  strictly  valid  generalization." 

The  Theorem  of  Le  Chatelier.  As  a  result  of  his  attempts  to 
modify  the  principle  of  maximum  work  and  render  it  generally 
applicable,  Le  Chatelier  was  led  to  the  discovery  of  a  rigorous  law 
of  wide-reaching  usefulness.  His  generalization  may  be  stated 
as  follows:  —  Any  alteration  in  the  factors  which  determine  an  equi- 
librium, causes  the  equilibrium  to  become  displaced  in  such  a  way 
as  to  oppose,  as  far  as  possible,  the  effect  of  the  alteration.  If  the 
temperature  of  a  system  which  is  in  equilibrium  be  raised,  or 
*  Essai  de  Mecanique  Chimique. 


THERMOCHEMISTRY  299 

lowered,  the  resulting  displacement  of  the  equilibrium  is  accom- 
panied by  such  absorption  or  evolution  of  heat,  as  will  tend  to 
maintain  the  temperature  constant.  An  interesting  illustration 
of  the  behavior  of  a  system  when  one  of  the  factors  controlling 
the  equilibrium  is  varied,  is  afforded  by  the  system, 

2  NO2  <=>  N2O4. 

The  reaction  proceeds  in  the  direction  indicated  by  the  upper 
arrow,  with  the  evolution  of  12.6  Cal.  Increase  of  temperature 
favors  the  reaction  which  is  accompanied  by  an  absorption  of 
heat,  which  in  this  case,  is  the  reaction  indicated  by  the  lower 
arrow.  Hence  as  the  temperature  rises,  the  percentage  of  N02 
increases  at  the  expense  of  N2O4.  This  fact  can  be  demonstrated 
by  the  following,  experiment:  Some  liquefied  N2O4  is  placed  in 
each  of  three  long  glass  tubes,  which  are  sealed  at  one  end.  When 
enough  N204  has  vaporized  to  displace  the  air,  the  open  ends  of 
the  tubes  are  sealed.  Changes  in  the  equilibrium,  caused  by  vary- 
ing the  temperature,  can  be  followed  by  noting  the  changes  in  the 
color  of  the  mixture.  The  gas,  N2O4  is  an  almost  colorless  sub- 
stance, while  NO2  is  reddish  brown.  At  ordinary  temperatures  the 
contents  of  the  tubes  will  be  brown  in  color.  One  tube  is  set 
aside  as  a  standard  of  comparison,  while  the  temperature  of  the 
second  is  lowered  by  surrounding  it  with  a  freezing  mixture.  As 
the  temperature  falls,  the  brown  color  of  the  contents  of  the  tube 
becomes  much  lighter,  showing  an  increased  formation  of  N204. 
The  third  tube  is  heated,  by  immersing  it  in  a  beaker  of  boiling 
water.  As  the  temperature  rises,  the  contents  of  the  tube  be- 
comes much  darker  in  color,  indicating  an  increase  in  the  amount 
of  NO2  in  the  mixture. 

Another  example  is  afforded  by  the  equilibrium  between  ozone 
and  oxygen,  represented  by  the  equation, 

2  O3  <=±  3  O2. 

The  reaction  indicated  by  the  upper  arrow  is  exothermic.  In- 
crease of  temperature  causes  a  displacement  of  the  equilibrium 
in  the  direction  of  the  lower  arrow,  since  under  these  conditions 
heat  is  absorbed.  Thus,  as  the  temperature  rises,  ozone  becomes 
increasingly  stable.  Nernst  has  calculated  that  at  6000°  C., 
the  temperature  of  the  photosphere  of  the  sun,  10  per  cent  of  the 


300  THEORETICAL  CHEMISTRY 

above  equilibrium  mixture  would  be  ozone.  Other  applications 
of  the  theorem  of  Le  Chatelier  will  be  given  in  subsequent 
chapters. 

REFERENCE 

Thermochemistry.     Thomsen.     Translated  by  Burke. 

PROBLEMS 

Yj__l}  From  the  following  data  calculate  the  heat  of  formation  of  HN02 
aq.— 

'  [NH4N02]  =  (N,)  +  2  H20  +  71.77  CaL, 
9  2  (H,)  +  (0,)  =  2  H20  +  136.72  CaL, 
?  (N«)  +  3  (H,)  +  aq.  =  2  NH3  aq.  +  40.64  CaL, 
;  NH3  aq.  +  HN02  aq.  =  NH4N02  aq.  +  9.110  CaL, 
'  [NH4N02]  +  aq.  =  NH4N02  aq.  -  4.75  CaL    • 

Ans.  (H)  +  (N)  +  (02)  +  aq.  =  HNO2  aq.  +  30.77  CaL 


2.  By  the  combustion  at  constant  pressure  of  2  grams  of  hydrogen 
with  oxygen  to  form  liquid  water  at  17°  C.,  68.36  CaL  are  evolved.  What 
is  the  heat  evolution  at  constant  volume?  Ans.  67.49  CaL 

>»(5J)  The  heats  of  solution  of  Na2S04,  Na2S04.H20,  and  Na2S04.10H20 
are  0.46,  —1.9  and  -18.76  CaL  respectively.  What  are  the  heats  of 
hydration  of  Na2S04;  (a)  to  monohydrate,  (b)  to  decahydrate? 

Ans.  (a)  2.36  CaL,  (b)  19.22  CaL 

4.   The  heats  of  neutralization  of  NaOH  and  NH4OH  by  HC1  are  13.68 
and  12.27  CaL  respectively.     What  is  the  heat  of  ionization  of  NH4OH, 
itis  assumed  to  be  practically  undissociated?  Ans.  —  1.41  CaL 

5v)From  the  following  energy  equations:  — 

/  [C]  +  (0,)  =  (CO,)  +  96.96  CaL, 
i  2  (H,)  +  (0,)  =  2  H20  +  136.72  CaL, 
o  2  C6H6  +  15  (0,)  =  12  (CO,)  -f  6  H2O  +  1598.7  CaL, 
'2  (C2H2)  +  5  (0,)  =  4  (CO,)  +  2  H20  +  620.1  CaL, 

all  at  17°  C.  and  constant  pressure,  calculate  the  heat  evolved  at  17°  C. 
in  the  reaction 

3  O2H2   = 


(a)  at  constant  pressure,  and  (b)'  at  constant  volume. 

Ans.  (a)  130.8  CaL,  (b)  129.07  CaL 


6.  Calculate  the  heat  of  formation  of  sulphur  trioxide  from  the  follow- 
ing energy  equations:  — 


THERMOCHEMISTRY  301 

[PbO]  +  [S]  +  3  (0)  =  [PbS04]  +  1655  K. 
i  [PbO]  +  H2S04.5  H20  =  [PbSO4]  +  6  H20  +  233  K. 
3  [S]  +  3  (0)  +  6  H2O  =  H2S04.5  H20  +  1422  K. 

[SO,]  +  6  H20  =  H2S04.5  H20  +  411  K. 

.  Ans.  [S]  +  3  (0)  =  [SO,]  +  1011  K. 

IT 
7.  What  is  the  heat  of  formation  of  a  very  dilute  solution  of  calcium 

chloride?     (See  table  on  p.  297.)  Ans.  187.6  Cal. 

(3}  Thomsen  has  shown  that  the  heat  of  solution  of  1  mol  of  H2S04 
in  n  mols  of  water  can  be  calculated  by  the  formula: 

17860  n 
Hs  = 


n  +  1.798 

Calculate  the  heat  of  solution  of  1  mol  of  H2S04  in  (a)  5  mols  of  water  and, 
(b)  in  10  mols  of  water.  How  many  calories  are  evolved  when  10  mols 
of  water  are  added  to  a  solution  containing  1  mol  of  acid  in  10  mols  of 
af? 

From  the  preceding  formula,  calculate  the  differential  heat  of  dilu- 
tion for  a  mixture  containing  1  mol  of  H2S04  and  5  mols  of  water. 

10.  The  specific  heats  of  the  following  solutions,  NaOH  +  100  H20, 
HC1  +  100  H20,  and  NaCl  +  201  H20,  are  0.968,  0.964  and  0.978  re- 
spectively.    The  heat  of  neutralization  of  first  two  solutions  at  10°  is 
14253  calories.     Calculate  the  heat  of  neutralization  at  20°. 

11.  Calculate  the  quantity  of  heat  required  to  raise  the  temperature 
of  Lmol  of  steam,  at  constant  pressure,  from  100°  to  200°. 

(12/) Calculate  the  number  of  calories  required  to  raise  the  temperature 
of  60  grams  of  nitric  oxide,  at  constant  volume,  from  100°  to  150°. 

13.  The  heat  of  combustion  of  ethyl  alcohol  is  341800  cals.,  and  the 
heats  of  formation  of  carbon  dioxide  and  water  are  96960  cals.,  and  68369 
cals.  respectively,  all  at  constant  pressure.  Calculate  the  heat  of  forma- 
tion of  ethyl  alcohol. 

(tJ)  Compute  the  molal  heat  of  combustion  of  trimethylmethane  by 
Thornton's  rule. 


CHAPTER  XII 
HOMOGENEOUS  EQUILIBRIUM 

Historical  Introduction.  In  this,  and  the  two  succeeding 
chapters,  the  conditions  which  affect  the  rate,  and  the  extent  of 
chemical  reactions  will  be  considered.  When  two  substances 
react  chemically,  it  is  customary  to  refer  the  phenomenon  to  the 
existence  of  an  attractive  force  known  as  chemical  affinity. 

Ever  since  the  metaphysical  speculations  of  the  Greeks,  who 
endowed  the  atoms  with  the  instincts  of  love  and  hate,  the  nature 
of  chemical  affinity  has  been  under  discussion.  Newton's  discov- 
ery of  the  law  of  gravitation  led  him  to  consider  the  attraction 
between  atoms,  and  the  attraction  between  large  masses  of  matter 
as  manifestations  of  the  same  force.  Although  Newton  found 
that  chemical  attraction  does  not  follow  the  law  of  the  inverse 
square,  yet  his  suggestion  exerted  a  profound  influence  upon  the 
minds  of  his  contemporaries. 

Geoffroy  and  Bergmann  arranged  chemical  substances  in  the 
order  of  their  displacing  power.  Thus,  if  we  have  three  sub- 
stances, Ay  B,  and  C,  and  the  attraction  between  A  and  B  is 
greater  than  that  between  A  and  C,  then  when  B  is  added  to  AC, 
it  will  completely  displace  C,  as  indicated  by  the  following  equa- 
tion: 

AC  +  B  =  AB  +  C. 

These  investigators  overlooked  a  factor  of  fundamental  import- 
ance in  conditioning  chemical  reactivity,  namely,  the  influence  of 
mass.  The  importance  of  the  relative  amounts  of  the  reacting 
substances  in  determining  the  course  of  a  reaction,  was  first  clearly 
recognized  by  Wenzel,*  in  1777.  It  remained  for  Berthollet,f 
however,  to  point  out  the  significance  of  the  views  advanced  by 
Wenzel.  His  first  paper  on  this  subject  was  published  in  1799, 
while  acting  as  a  scientific  adviser  to  Napoleon  on  his  Egyptian 

*  Lehre  von  der  chemischen  Verwandtschaft  der  Korper. 
t  Essai  de  Statique  Chimique. 
302 


HOMOGENEOUS  EQUILIBRIUM  303 

expedition.     Under  ordinary  conditions,  sodium   carbonate  and 
calcium  chloride  react  according  to  the  equation, 

+  CaCl2  =  2  NaCl  +  CaCO3. 


This  reaction  was  found  to  proceed  nearly  to  completion. 
Berthollet  observed  the  deposits  of  sodium  carbonate  on  the  shores 
of  certain  saline  lakes  in  Egypt,  and  pointed  out,  that  this  salt  is 
produced  by  the  reversal  of  the  above  reaction,  the  large  excess 
of  sodium  chloride  in  solution  in  the  water  of  the  lakes  condi- 
tioning the  course  of  the  reaction. 

The  German  chemist,  Rose,*  furnished  much  additional  evidence 
in  favor  of  the  effect  of  mass  on  chemical  reactions.  He  pointed 
out,  that  in  nature,  the  silicates,  which  are  among  the  most  stable 
compounds  known,  are  undergoing  a  continual  decomposition 
under  the  influence  of  such  relatively  weak  agents  as  water  and 
carbon  dioxide.  The  relatively  strong  specific  affinities  of  the 
atoms  of  the  silicates,  are  overcome  by  the  preponderating  masses 
of  water  and  carbon  dioxide  in  the  atmosphere. 

In  1862,  an  important  contribution  to  our  knowledge  of  the 
effect  of  mass  on  the  course  of  a  chemical  reaction  was  made  by 
Berthelot  and  Pean  de  St.  Gilles.  f  They  investigated  the  forma- 
tion of  esters  from  alcohols  and  acids.  The  reaction  between 
ethyl  alcohol  and  acetic  acid  is  represented  by  the  equation, 

C2H5OH  +  CHsCOOH  <=±  CH3COOC2H5  +  H2O. 

Starting  with  equivalent  quantities  of  alcohol  and  acid,  the  reac- 
tion proceeds,  until  about  two-thirds  of  the  reacting  substances 
have  been  converted  into  ester  and  water.  In  like  manner,  if 
equivalent  quantities  of  ethyl  acetate  and  water  are  brought 
together,  the  reaction  proceeds  in  the  direction  indicated  by  the 
lower  arrow,  until  about  one-third  of  the  original  substances  have 
been  converted  into  acid  and  alcohol.  In  other  words,  the  reac- 
tion is  reversible,  a  condition  of  equilibrium  resulting,  when  the 
speeds  of  the  two  reactions,  indicated  by  the  upper  and  lower 
arrows,  become  equal.  If  now  a  fixed  amount  of  acid  is  taken, 
say  1  equivalent,  and  the  quantity  of  alcohol  is  varied,  a  corre- 
sponding displacement  of  the  equilibrium  follows.  The  following 
table  gives  the  results,  obtained  by  Berthelot  and  Pean  de  St. 

*  Pogg.  Ann.,  94,  481  (1855);  95,  96,  284,  426  (1855). 

t  Ann.  Chim.  Phys.  [3],  65,  385;  66,  5;  68,  225  (1862-1863). 


304 


THEORETICAL  CHEMISTRY 


Gilles,  for  ethyl  alcohol  and  acetic  acid.  The  first  and  third 
columns  give  the  number  of  equivalents  of  alcohol  to  1  equiva- 
lent of  acetic  acid,  and  the  second  and  fourth  columns  give  the 
percentage  of  ester  formed. 


EQUILIBRIUM  IN   REACTION   BETWEEN  ACETIC 
ACID  AND  ALCOHOL 


Equivalents 

Eater 

Equivalents 

Ester 

of  Alcohol. 

Formed. 

of  Alcohol. 

Formed. 

0.2 

19.3 

2.0 

82.8 

0.5 

42.0 

4.0 

88.2 

1.0 

66.5 

12.0 

93.2 

15 

77.9 

50.0 

100.0 

The  effect  of  increasing  the  mass  of  alcohol  on  the  course  of  the 
reaction  is  very  beautifully  shown  by  the  above  results. 

The  Law  of  Mass  Action.  While  the  influence  of  the  relative 
masses  of  the  reacting  substances  in  conditioning  chemical  reac- 
tions was  thus  fully  established,  it  was  not  until  1867  that  the  law 
governing  the  action  of  mass  was  accurately  formulated. 

In  that  year,  Guldberg  and  Waage,*  two  Scandinavian  investiga- 
tors, enunciated  the  law  of  mass  action  as  follows:  —  The  rate,  or 
speed,  of  a  chemical  reaction  is  proportional  to  the  active  masses  of 
the  reacting  substances  present  at  that  time.  Guldberg  and  Waa.ge 
defined  the  term  "  active  mass,"  as  the  molecular  concentration 
of  the  reacting  substances.  It  is  to  be  carefully  noted,  that  the 
amount  of  chemical  action  is  not  proportional  to  the  actual 
masses  of  the  substances  present,  but  rather  to  the  amounts 
present  in  unit  volume.  The  law  is  generally  applicable  to  homo- 
geneous systems;  that  is,  to  those  systems  in  which  ordinary 
observation  fails  to  reveal  the  presence  of  essentially  different 
parts.  The  amount  of  chemical  action  exerted  by  a  substance, 
can  be  determined,  either  from  its  effect  on  the  equilibrium,  or 
from  its  influence  on  the  speed  of  reaction. 

In  order  to  apply  the  law  of  mass  action  practically,  it  must  be 
formulated  mathematically.  Let  a  and  b  denote  the  molecular 
concentrations  of  the  substances  initially  present  in  a  reversible 
reaction.  According  to  the  law  of  mass  action,  the  rate  at  which 

*  Etudes  sur  les  Affinites  Chimiques,  Jour,  prakt.  Chem.  [2],  19,  69  (1879). 


HOMOGENEOUS  EQUILIBRIUM  305 

these  substances  combine  is  proportional  to  the  active  masses 
of  each  constituent,  and  therefore,  to  their  product,  ab.  The 
initial  speed  of  the  reaction,  at  the  time  to,  is  therefore, 

Speed0  °°  ab,  or  Speed/0  =  k  •  ab, 

in  which  the  proportionality  factor  k,  is  known  as  the  velocity 
constant.  As  the  reaction  proceeds,  the  molecular  concentrations 
of  the  original  substances  steadily  diminish,  while  the  molecular 
concentrations  of  the  products  of  the  reaction  steadily  increase. 
Let  us  assume,  that  after  the  interval  of  time  t,  x  equivalents  of 
the  products  of  the  reaction  have  been  formed.  The  speed  of 
the  original  reaction  will  now  be, 

Speed  =  k  (a  —  x)  (b  —  x). 

As  the  reaction  proceeds,  the  tendency  of  the  products  to  combine, 
and  reform  the  original  substances,  increases.  At  the  time  t, 
when  the  concentration  of  the  products  is  x,  the  speed  of  the 
reverse  reaction  will  be, 

Speed  =  ki  •  xz, 

where  k\  is  the  velocity  constant  of  the  reverse  reaction. 

We  thus  have  two  reactions  proceeding  in  opposite  directions; 
the  speed  of  the  direct  reaction  continuously  diminishes  while 
that  of  the  reverse  reaction  continually  increases.  It  is  evident 
that  a  point  must  ultimately  be  reached  at  which  the  speeds  of 
the  direct  and  reverse  reactions  become  equal,  and  a  condition 
of  dynamic  equilibrium  will  be  established.  Let  x\  represent  the 
value  of  x,  when  equilibrium  is  attained;  we  have  then, 

Speed  direct  =  k  (a,  -  Xi)  (b  -  Xi)   =  Speed  reverse  =    kiXi2, 

or 

(a  -xi)  (b  -Xi)_ki_Vr  m 

~~x?~          =  k  = 

in  which  K  is  known  as  the  equilibrium  constant.  Since  the  veloc- 
ity constants,  ft  and  ki,  are  independent  of  the  concentration,  it 
follows,  that  the  above  equation  holds  for  all  concentrations. 
Therefore,  if  the  value  of  the  equilibrium  constant  of  a  reaction 
is  known,  the  equilibrium  conditions  can  be  calculated  for  any 
concentrations  of  the  reacting  substances. 

When  more  than  one  mol  of  a  substance  is  involved  in  a  re- 


306  THEORETICAL  CHEMISTRY 

action,  each  mol  must  be  considered  separately  in  the  mass  action 
equation. 
Thus,  let 

+  ^Ag  +  .  .  .  ^  ni'Ai'  +  n*'Az'  +  .  . 


represent  any  reversible  reaction,  in  which  n\  mols  of  AI  and  712 
mols  of  A2,  react  to  form  n\  mols  of  A  /  and  n%  mols  of  A2'.  When 
equilibrium  is  attained,  we  shall  have, 


or 


CAl'CA2' 


in  which  the  symbol  c  is  used  to  denote  the  active  mass,  or  molecu- 
lar concentration,  of  the  substances  involved  in  the  reaction.  This 
is  a  perfectly  general  form  of  the  mass-action  equation.  Since  at 
any  one  temperature,  concentration  and  pressure  are  proportional, 
we  may  write  equation  (2)  in  the  following  form: 


v  /'Q^ 

=  Ap  •  (o) 

In  the  case  of  gaseous  equilibria,  this  is  often  a  more  convenient 
form  of  the  equation. 

The  relation  between  the  two  equilibrium  constants,  Kc  and  Kp, 

can  be  easily  determined,  as  follows :  —  Since  c  =  -  =  ~—    we 

v      til 

have,  on  substituting  this  value  of  c  in  equation  (2).    * 

(4) 


RTf     \RTl 

or,  indicating  the  sum  or  the  initial  number  of  mols  by  2n,  and  the 
sum  of  the  final  number  of  mols  by  Zn',  we  have, 

If  V     (  T?  TASn'  —  Sn  /K"\ 

Ac  =  Ap  (n,l)*n     *».  (5) 

It  is  evident,  therefore,  in  reactions  where  the  same  number  of  mols 
occur  on  both  sides  of  the  equality  sign,  that  Kc  =  Kv.  Equation 
(2)  is  sometimes  known  as  the  reaction  isotherm. 


HOMOGENEOUS  EQUILIBRIUM  307 

Derivation  of  the  Reaction  Isotherm.  Let  us  consider  a  system, 
consisting,  either  of  a  dilute  mixture  of  gases,  or  a  dilute  solution' 
contained  in  a  closed  vessel.  Let  us  assume,  that  the  equilibrium 
m  represented  by  the  equation, 

+  n2A2  +  .  .  .  +±  n/A/  +  nz'Azf      .  .  . 


has  been  established,  and  that  the  partial  pressures  of  AI,  A2,  AI 
and  A2',  are  pi,  p2  pi  and  p2',  respectively.  We  will  suppose 
that  n  molecules  of  the  reacting  substances  can  be  introduced  into 
the  reaction  chamber,  and  that  at  the  same  time,  n'  molecules  of 
the  products  of  the  reaction  can  be  removed  from  the  reaction 
chamber,  without  altering  the  equilibrium.  In  other  words,  for 
every  molecule  of  a  given  molecular  species  which  is  introduced 
into  the  system,  an  equivalent  number  of  molecules  of  another 
molecular  species  will  be  displaced,  so  that  the  change  in  the  re- 
action mixture  will  proceed  uniformly,  from  left  to  right,  with- 
out any  change  occurring  in  the  relative  concentrations.  Under 
these  conditions,  the  reaction  can  be  conducted  isothermally  and 
reversibly,  and  the  work  performed  can  be  calculated  thermody- 
namically.  As  has  been  shown  in  the  chapter  on  thermody- 
namics, (p.  134)  the  work  required  to  introduce  n\  mols  of  A\ 
into  the  reaction  chamber  against  its  partial  pressure,  'pit*  is 
niRTloge  PI/PQ,  where  pQ  is  the  normal  external  pressure. 

In  like  manner,  the  corresponding  amount  of  work  required 
to  introduce  n2  mols  of  A2  into  the  reaction  chamber,  will  be 
n2R  T  loge  p2/po-  Therefore,  the  total  work  required  to  introduce 
n\  mols  of  AI,  and  n2  mols  of  Az,  into  the  reaction  chamber  will  be, 

wi  =  -  R  T   m  loge      +  nsX  log,  (6) 


the  negative  sign  indicating  work  done  on  the  system. 

Similarly,  the  total  work  performed  by  the  system,  when  n\ 
mols  of  A/,  and  n2f  mols  of  A2',  are  withdrawn  from  the  reaction 
chamber,  will  be 

w2  =  RT  (n,'  log,  &  +  ng'  logX-  (7) 


The  sum  of  equations  (6)  and  (7)  will  give  the  total  gain  in  work, 
resulting  from  the  isothermal  transformation  of  n\  mols  of  A\,  and 

*  It  should  be  remembered  that  pi,  p2  etc.,  may  be  either  gaseous,  or  osmotic 
pressures. 


308  THEORETICAL  CHEMISTRY 

n%  mols  of  A%,  into  n\  mols  of  A  \  and  n^',  mols  of  A%'.  Performing 
this  addition,  and  rearranging  terms,  we  have 

wi  +  wz  =  W  =  RT  (HI  log,  pQ  +  nz  loge  pQ  -  HI  loge  p0  -  n*'  loge  p0) 
+  RT  (ni  loge  pi  -f  n*'  log*  p2'  -  HI  loge  pi  -  nz  loge  p2).      (8) 

Since  the  value  of  the  external  pressure,  p0,  has  nothing  to  do  with 
the  values  of  the  partial  pressures  of  the  constituents  of  the  reac- 
tion mixture  at  equilibrium,  it  may  be  placed  equal  to  unity. 
All  terms  involving  loge  p0,  will  then  become  zero,  and  equation 
(8)  may  be  written  in  the  following  form : 

W  =  RT  (nif  loge  Pl    +  Wg'  loge  ps'  -  Hi  loge  Pi  ~  **  loge  PS).      (9) 

The  value  of  W  given  in  equation  (9),  it  should  be  observed,  is 
the  maximum  work  which  the  reaction  can  perform,  and  hence,  is 
constant,  provided  the  temperature  remains  unchanged. 

On  rearranging  equation  (9),  we  have 

W  =  flTloge^y  =  constant,  (10) 

and,  since  the  logarithm  of  a  constant  is  also  a  constant,  it  follows 
that  equation  (10)  reduces  to  the  same  form  as  that  of  equation 
(3),  namely, 

7)i  '"1*7)0 'W2/ 

Kv=plnp\    ,  (11) 

*  «     Pinip2n2 

which  expresses  the  law  of  mass  action  in  its  most  general  form. 
A  simpler  kinetic  derivation  of  the  reaction  isotherm,  has 
been  furnished  by  van't  Hoff.  If  we  assume,  that  the  rate  of 
chemical  change  is  proportional  to  the  number  of  collisions  per 
unit  of  time  between  the  molecules  of  the  reacting  substances, 
then  in  the  reaction, 

niAi  4-  nzAz  +    .  .  .  =  n/Ai'  +  nz'A2'  +    .  .  .  , 
the  velocity  of  the  direct  change  will  be  kcn^  c*%  .  .  . ,  and  the 
velocity  of  the  reverse  reaction  will  be  fe^iA.  •  •  • 
At  equilibrium," the  two  velocities  will  be  equal,  and  therefore, 


or 


--.=       K 

...      fe 


HOMOGENEOUS  EQUILIBRIUM  309 

As  a  consequence  of  the  assumptions  involved  in  both  the  thermo- 
dynamic  and  the  kinetic  proofs  of  the  law  of  mass  action,  it  fol- 
lows, that  the  law  is  only  strictly  applicable  to  very  dilute  solutions. 
Notwithstanding  this  limitation,  experimental  results  indicate 
that  it  frequently  holds  for  moderately-concentrated  solutions. 

Equilibrium  in  Homogeneous  Gaseous  Systems. 

(a)  Decomposition  of  Hydriodic  Add.  A  typical  example  of 
equilibrium  in  a  gaseous  system  is  afforded  by  the  decomposition 
of  hydriodic  acid,  as  represented  by  the  equation, 

H2  +  I2  <±  2  HI. 

This  reaction  has  been  thoroughly  investigated  by  Hautefeuille, 
Lemoine  and  Bodenstein.*  The  reaction  is  well  adapted  for 
investigation,  since  it  proceeds  very  slowly  at  ordinary  tempera- 
tures, while  at  the  temperature  of  boiling  sulphur,  448°  C.,  equi- 
librium is  established  quite  rapidly.  If  the  mixture  of  gases 
is  maintained  at  448°  C.  for  some  time,  and  is  then  cooled  quickly, 
the  respective  concentrations  of  the  components  of  the  mixture 
can  be  determined  by  the  ordinary  methods  of  chemical  analysis. 
Various  mixtures  of  the  gases  are  sealed  in  glass  tubes,  and  heated 
for  a  definite  time  in  the  vapor  of  boiling  sulphur.  The  tubes 
are  then  cooled  rapidly  to  the  temperature  of  the  room,  and  after 
the  iodine  and  hydriodic  acid  have  been  removed  by  absorption  in 
potassium  hydroxide,  the  amount  of  free  hydrogen  present  in  each 
tube  is  measured. 
Applying  the  law  of  mass  action  to  the  above  equation,  we  have 

Kc. 

Expressing  the  analytical  results  in  mols,  let  a  mols  of  iodine  be 
mixed  with  6  mols  of  hydrogen,  and  let  2  x  mols  of  hydriodic  acid 
be  formed.  Then,  when  equilibrium  is  established,  a  —  x  will  be 
the  amount  of  iodine  vapor,  and  b  —  x  will  be  the  amount  of  hydro- 
gen present.  The  concentrations  being  directly  proportional  to 
the  amounts  present,  we  may  substitute  these  values  for  c#2,  c/2, 
and  CHI  in  the  mass-action  equation.  The  following  expression 
is  thus  obtained: 

(6  -  x  )  (a  -  x)  _  „ 


Zeit.  Phys.,  Chem.,  22,  1  (1897). 


310 


THEORETICAL  CHEMISTRY 


Solving  the  equation  for  x,  we  obtain, 


a  +  6  -  V  a2  +  62  -  ab  (2  -  16  Ke) 
1-4KC 

Since,  according  to  Avogadro's  hypothesis,  equal  volumes  of  all 
gases  contain  the  same  number  of  molecules,  volumes  may  be 
substituted  for  a,  6,  and  x.  Bodenstein  expressed  his  results  in 
terms  of  volumes,  reduced  to  standard  conditions  of  temperature 
and  pressure.  On  analyzing  equilibrium  mixtures,  Bodenstein 
found,  that  at  448°  C.,  Kc  =  0.01984,  and  at  350°  C.,  Kc  =  0.01494. 
Having  determined  the  value  of  the  equilibrium  constant,  he 
made  use  of  this  value  to  calculate  the  volume  of  hydriodic 
acid  which  should  be  obtained  from  known  volumes  of  hydrogen 
and  iodine.  A  comparison  of  the  calculated  and  observed  values, 
showed  excellent  agreement.  The  following  table  contains  a  few 
of  the  results  obtained  by  Bodenstein  at  448°  C. 

EQUILIBRIUM  IN  REACTION   BETWEEN  HYDROGEN 
AND  IODINE 


Hydrogen, 
b. 

Iodine, 
a. 

HI  calculated, 
2x. 

HI  observed, 
2x. 

20.57 

5.22 

10.19 

10.22 

20.60 

14.45 

25.54 

25.72 

15.75 

11.90 

20.65 

20.70 

14.47 

38.93 

27.77 

27.64 

8.10 

2.94 

5.64 

5.66 

8.07 

9.27 

13.47 

13.34 

It  is  of  interest  to  note  that  a  change  in  pressure  does  not 
alter  the  equilibrium  in  this  gaseous  system.  Making  use  of  the 
partial  pressures  of  the  components  of  the  gaseous  system,  instead 
of  the  concentrations,  we  have 

PHZ  •  PA  =  K 

Now  let  the  total  pressure  on  the  system  be  increased  to  n  times 
its  original  value;  under  these  conditions  the  partial  pressures 
are  all  increased  in  the  same  proportion,  and  we  have 


•  npit  _ 
*mi  ' 


^PJ 


HOMOGENEOUS  EQUILIBRIUM  311 

which  will  be  seen  to  be  equivalent  to  the  original  expression, 
since  n  cancels  out.  The  equilibrium  is  thus  seen  to  be  indepen- 
dent .aLthe. pressure.  This  is  only  true  for  those  systems  in  which 
a  change  in  volume  does  not  occur. 

(6)  Dissociation  of  Phosphorus  Pentachloride.  When  phos- 
phorus pentachloride  is  vaporized  it  dissociates  according  to  the 
following  equation, 

PC15  <^  PCla  +  C12. 


Applying  the  law  of  mass  action,  we  have 

2          JT 
=  Kc. 


Starting  with  1  mol  of  phosphorus  pentachloride,  whichi  if  undis- 
sociated  would  occupy  the  volume  V,  under  atmospheric  pressure, 
and  letting  a  denote  the  degree  of  dissociation,  the  molecular  con- 
centrations at  equilibrium  will  be  as  follows: 

1  —  a  a.  ,  a 

Letting  (1  +  a)  V  =  V,  and  substituting  in  the  above  equation, 
we  have 


-x- 
V 


-K 

a)V-K" 


At  250°  C.,  phosphorus  pentachloride  is  dissociated  to  the  extent 
of  80  per  cent.     Under  atmospheric  pressure,  1  mol  will  be  pres- 

273  -I-  250 
ent  in,  22.4  —  _     c-  liters  =  V.    Therefore,  the  final  volume 


will  be, 

273  +  250\ 
273      / 


V=(l  +  0.8)  (22.4 


The  value  of  the  equilibrium  constant  —  usually  designated  ii 
cases  of  dissociation,  the  dissociation  constant  —  is,  therefore, 

R  =_ (0.8)2 

(1  -  0.8)  (1  +  0.8)  (  22.4  2732+325°) 


312  THEORETICAL  CHEMISTRY 

Having  obtained  the  value  of  Kc,  the  direction  and  extent 
of  the  reaction  at  250°  C.  can  be  determined,  provided  the 
initial  molecular  concentrations  are  known.  The  reaction  is 
accompanied  by  a  change  in  volume,  and  therefore,  the 
equilibrium  is  displaced  by  a  change  in  pressure.  Making 
use  of  the  partial  pressures  of  the  components  of  the  gaseous 
mixture,  we  have, 

&-K 
p,  =    *" 

where  pi  and  p2,  are  the  partial  pressures  of  phosphorus  penta- 
chloride  and  the  products  of  the  dissociation,  phosphorus  tri- 
chloride and  chlorine,  respectively.  Let  the  total  pressure  be 
increased  n-times,  then, 

g    _  n2p22  _  npi* 
npi  "pi' 

It  is  apparent  from  this  equation,  that  the  equilibrium  is  not 
independent  of  the  pressure,  an  increase  in  pressure  being 
accompanied  by  a  diminution  of  the  dissociation.  An  import- 
ant point  in  connection  with  dissociation,  first  observed  by 
Deville,*  is  the  effect  on  the  equilibrium  of  the  addition  of 
an  excess  of  one  of  the  products  of  dissociation.  For  example, 
in  the  equilibrium, 

*  PC13  +  C12, 


an  excess  of  chlorine,  or  of  phosphorus  trichloride,  drives  back  the 
dissociation.  If  p\  denotes  the  partial  pressure  of  phosphorus 
pentachloride,  pz  that  of  phosphorus  trichloride,  and  p3  that  of 
chlorine,  then  we  have, 


K,. 

Pi 

Now  let  an  excess  of  chlorine  be  added;  this  will  cause  the  value 
of  ps  to  increase.  Since  the  value  of  Kp  is  constant,  the  value  of 
pz  must  dimmish,  and  that  of  pi  must  increase.  Hence,  the  addi- 
tion of  an  excess  of  either  product  of  dissociation  causes  a  diminu- 
tion of  the  amount  of  the  dissociation. 

*  Lemons  sur  la  dissociation,  Paris  (1866). 


HOMOGENEOUS  EQUILIBRIUM  313 

(c)  Dissociation  of  Carbon  Dioxide.     Carbon  dioxide  dissociates 
according  to  the  equation, 


This  is  a  somewhat  more  complex  gaseous  system  than  either  of 
the  foregoing  systems.  When  equilibrium  is  established,  let 
pi  be  the  partial  pressure  of  the  carbon  dioxide,  p2  the  partial  pres- 
sure of  carbon  monoxide,  and  p^  the  partial  pressure  of  oxygen, 
then  we  have 


Pi2 

At  3000°  C.,  and  under  atmospheric  pressure,  carbon  dioxide  is 
40  per  cent  dissociated.  The  partial  pressures  of  each  of  the  com- 
ponents may  be  readily  calculated  as  follows: 

2(1-0.40) 
2  (1  -  0.40)  +  3  X  0.40 

2  X  0.40 
2(1  -0.40)  +3  X0.40 

0.40 

2  (1  -  0.40)  +  3  X  0.40 

Substituting  these  values  in  the  above  equation,  we  obtain, 

(0.33)*X0.17 
p  (0.50)2 

The  dissociation  constant  for  carbon  dioxide  may  have  a  different 
value  if  the  equation  is  written  in  the  form, 

C02  4=*  CO  +  }  O2. 
Applying  the  law  of  mass  action,  we  have, 

K,=l 

ifl 

Substituting  the  above  values  of  the  partial  pressures,  we 
obtain, 

Kp  =  0.272. 

Equilibrium   in   Liquid   Systems.     The   reaction   between   an 
alcohol  and  an  acid,  to  form  an  ester  and  water,  may  be  taken  as 


_     -          r        _     -         r         _    _  m        ,    r 

ale.    ~  '      v»  acid  -  >     V  ester    ~         j    HUU     O  water 


314  THEORETICAL  CHEMISTRY 

an  example  of  equilibrium  in  a  liquid  system.  In  the  reac- 
tion, 

C2H5OH  +  CH3COOH  <=>  CH3COOC2H5  +  H2O, 

let  a,  b,  and  c  represent  the  number  of  mols  of  alcohol,  acid  and 
water  respectively,  which  are  present  in  V  liters  of  the  mixture, 
and  let  x  denote  the  number  of  mols  of  ester  and  water  which 
have  been  formed,  when  the  system  has  reached  equilibrium. 
The  active  masses  of  the  components  will  be, 

a-  x  b  -  x  x_ 

y 

Applying  the  law  of  mass  action,  we  obtain  , 

v  (c  +  a?)  ~ 

(a-x}(b-x]  ~~ 

In  this  case,  the  value  of  the  equilibrium  constant  is  independent 
of  the  volume.  This  reaction  has  been  studied,  as  already  men- 
tioned, by  Berthelot  and  Pean  de  St.  Gilles.*  They  found  that 
when  equivalent  amounts  of  alcohol  and  acid  are  mixed,  the  reac- 
tion proceeds  until  two-thirds  of  the  mixture  is  changed  into  ester 
and  water.  Hence,  we  find, 

K  -  i  x*  -  4 

Kc  ~  *  x  J  - 

Having  determined  the  value  of  Kc,  it  may  now  be  used  to  cal- 
culate the  equilibrium  conditions  for  any  initial  concentrations 
of  the  substances  involved  in  the  reaction.  As  an  illustration, 
we  will  take  1  mol  of  acetic  acid  and  treat  it  with  varying  amounts 
of  alcohol,  the  initial  mixture  containing  neither  of  the  products  of 
the  reaction.  The  equation  takes  the  form, 


=  4. 


(a  -x)(l-  x) 
Solving  for  x,  we  have, 

x  =  f  (1  +  a  -  Va?  -  a  -f  1). 

A  comparison  of  the  observed  and  calculated  values,  given  in  the 
accompanying  table,  shows  that  the  agreement  is  excellent,  even 
in  the  more  concentrated  solutions,  where  we  might  reasonably 
expect  that  the  mass  law  would  cease  to  hold. 

*  Loc.  cit. 


HOMOGENEOUS  EQUILIBRIUM 


315 


EQUILIBRIUM  IN  THE  REACTION  BETWEEN  ACETIC 
ACID  AND  ALCOHOL 


Alcohol, 
a. 

Ester 
(observed), 

X. 

Ester 
(calculated), 

X. 

0.05 

0.05 

0.049 

0.08 

0.078 

0.078 

0.18 

0.171 

0.171 

0.28 

0.226 

0.232 

0.33 

0.293 

0.311 

0.50 

0.414 

0.523 

0.67 

0.519 

0.528 

1.0 

0.665 

0.667 

1.5 

0.819 

0.785 

2.0 

0.858 

0.845 

2.24 

0.876 

0.864 

8.0 

0.966 

0.945 

Derivation  of  the  Reaction  Isochore.  The  displacement  of  the 
equilibrium  of  a  system  at  constant  volume,  resulting  from  a  change 
in  temperature,  may  readily  be  determined  by  applying  the  Gibbs- 
Helmholtz  equation  (see  p.  139)  to  the  equation  of  the  reaction 
isotherm.  Since  at  the  same  temperature,  concentrations  are  pro- 
portional to  partial  pressures,  equation  (10)  may  be  written  in  the 
form, 

W  =  RT  loge  Vr!,?*"2  =  constant.  (12) 


But,  according  to  equation  (2), 


_    „ 


(13) 


and  therefore,  we  may  write, 

W  =  RT\ogeKc. 
Differentiating  equation  (13),  we  have 

dW  =  RdT  loge Kc  +  RTd  (log*  Kc),  (14) 

which,  when  combined  with  the  Gibbs-Helmholtz  equation,  viz., 


W+U-T™. 


gives 


U 

dT         '  RT2 


(15) 


316  THEORETICAL  CHEMISTRY 

Since  the  process  takes  place  at  constant  volume,  it  follows  that  the 
decrease  in  total  energy  U,  may  be  replaced  by  —  QVt  the  heat 
evolved  when  no  external  work  is  performed.  Therefore  equation 
(15)  becomes, 

-  Q, 


This  expression  which  was  originally  derived  by  van't  Hoff,  is 
commonly  known  as"  the  equation  of  the  reaction  isochore.  It 
should  be  noted  that  in  the  derivation  of  equation  (16),  we  are 
dealing  with  the  concentrations,  and  not  with  the  partial  pressures, 
of  the  different  molecular  species  involved  in  the  reaction.  While 
it  was  immaterial  in  the  equation  of  the  reaction  isotherm  whether 
we  made  use  of  concentrations,  or  pressures,  this  is  not  the  case 
with  the  reaction  isochore,  since  at  constant  volume,  the  con- 
centration of  a  substance  remains  constant  while  the  pressure  does 
not. 

In  order  to  derive  a  similar  expression,  involving  Kp  and  Qv, 
we  make  use  of  equation  (5),  expressing  the  relation  between  the 
two  equilibrium  constants,  viz., 

Kc 


On  substituting  this  value  of  Kc  in  equation  (16),  we  have 

'-Sn'#*]  =  ^r>  (17) 

which  becomes  on  simplifying, 
-^=,  floge  Kp  +  loge 


or, 

d  f~i        ^    •   (Sn  ~  Sn/)  dTl  _  ~  Q* 

T7S     lOge   Aj,  H ^ —     Ttm* 

dl   L  ^  J  /fc^ 

Therefore, 

d  (loge  K9)       -  Qv  _  /Sn  -  SrA 
dT7         =  ^T72      V       T       /' 

-  (Sn  -  2n')  fi T  -  QP 
/^T2 


(18) 


The  term  —  (Sn  —  Sw')  RT,  however,  is  equivalent  to  the  ex- 
ternal work  nRT;  but 

nRT  =  Q0-  Qp;    (see  p.  286), 


HOMOGENEOUS  EQUILIBRIUM  317 

hence,  equation  (18)  becomes, 

d  (log.  Kp)  _-Qp 
dT          '  RT2  ' 

Both  equations  (16)  and  (19)  show  that  the  rate  of  change  of 
the  natural  logarithm  of  the  equilibrium  constant  with  tempera- 
ture, is  equal  to  the  total  heat  of  reaction,  divided  by  the 
molar  gas  constant  multiplied  by  the  square  of  the  absolute 
temperature  at  which  the  reaction  takes  place. 

Equations  (16)  and  (19)  hold  only  for  displacements  of  the 
equilibrium  due  to  infinitely  small  changes  in  temperature.  In 
order  to  render  these  equations  applicable  to  concrete  equilibria, 
it  is  necessary  to  integrate  them.  The  integration  of  these  ex- 
pressions can  only  be  performed  if  Q  is  constant.  For  small  inter- 
vals of  temperature,  Q  is  practically  constant,  and  for  larger  in- 
tervals, we  may  take  the  value  of  Q  which  corresponds  to  the 
mean  of  the  two  temperatures  between  which  the  integration 
is  performed.  Integrating  equations  (16)  and  (19)  on  this  as- 
sumption, we  obtain, 


log,  ^  -  log,    JT.,   =    -  -      r      ,  (20) 

and 

log,  KK  -  log,  K»  =  =e    ±  -  1    -  (21) 


Passing  to  Briggsian  logarithms,  and  putting  R  =  1.99  calories, 
equations  (20)  and  (21)  become, 


(22) 
and 

(23) 


The  assumption  that  Qv  and  QP,  in  equations  (20)  and  (21),  are 
independent  of  the  temperature  is,  in  fact,  only  approximately 
true,  even  for  relatively  small  temperature  intervals.  It  has  been 
pointed  out  by  Nernst  and  others,  that  the  total  energy  C7,  may 
be  expressed  in  terms  of  a  series  involving  ascending  powers  of  T. 
Thus  the  change  in  the  total  energy  of  a  gaseous  reaction  with 
the  temperature,  may  be  expressed  by  the  series, 

U  =  U0  +  aT  +  0T*  +  .  .  ., 


318  THEORETICAL  CHEMISTRY 

in  which  UQ  is  the  heat  of  the  reaction  in  the  vicinity  of  the  ab- 
solute zero.  Introducing  this  value  of  U  into  equation  (15)  and 
integrating,  we  have 

Kc  =         -      log,  T  -       T  -  .  .  .  +  i,          (24) 


where  i  is  the  constant  of  integration.  If  the  integration  be 
carried  out  between  any  two  temperatures,  T\  and  T2,  the  in- 
tegration constant  disappears,  and  we  obtain  the  expression, 

-^  (jr  -  JT)  -  flog,  ^  -  |  (Tt-  7,).  (25) 


For  further  information  as  to  the  significance  of  equation  (25),  the 
student  must  familiarize  himself  with  the  Nernst  heat  theorem. 
We  shall  now  proceed  to  show  how  these  important  equations 
may  be  applied  to  several  typical  equilibria. 

(a)  Vaporization  of  Water.     The  equilibrium  between  a  liquid 
and  its  vapor  is  conditioned  by  the  pressure  of  the  vapor,  this 
in  turn  being  dependent  upon  the  temperature.     In  this  case 
we  have  KPI  =  pi,  and  KP2  =  p2.     The  value  of  Qv  for  water  can 
be  calculated  from  the  following  data: 

Ti  =  273°,  pi  =    4.54  mm.  of  mercury, 

T2  =  273°  +  11°.54,  p2  =  10.02  mm.  of  mercury. 

Substituting  in  equation  (23),  we  have 

=  4.581  (log  10.02  -  log  4.54)  273  X  284.5 
"  ^v  ~  284.5  -  273 

or 

-  Qp  =  -  10,670  calories. 

The  value  of  Qp  obtained  by  experiment  is  —  10,854  calories. 

(b)  Dissociation  of  Nitrogen  Tetroxide.     In  the  reaction, 

N2O4  ?=*  2  NO2, 

the  following  values  for  the  dissociation  of  N2O4  have  been  ob- 
tained : 

Ti  =  273°  +    26°.  1,  ai  =  0.1986, 

Tz  =  273°  +  111°.3,  «2  =  0.9267. 


HOMOGENEOUS  EQUILIBRIUM  319 

If  the  dissociation  takes  place  under  a  pressure  of  1  atmosphere, 
then  the  partial  pressures  of  the  component  gases  will  be 

~  and 


The  values  of  KPI  and  K&  are,  then,  according  to  the  law  of 
mass  action  as  follows: 

2«i 

—  I 

4«!2 


and 


2  V 


-|-«2/ 

?"T=3T 


Substituting  in  equation  (23)  and  solving  for  Qp,  we  obtain 

4  581  Flog  4  X  (°'9267)2     log4  x  (Q-1986)2] 
_0  M-(0.9267)2~i0gl-(0.1986)2J^ 

384.3  -  299.1 
or, 

Qp  =  -  12,260  calories  per  mol  of  N2O4. 

In  a  reaction  which  is  accompanied  by  no  thermal  change, 
Qv  =  Qp  =  0,  and  the  right-hand  side  of  equations  (16)  and  (19) 
becomes  equal  to  zero.  In  other  words,  in  such  a  reaction  a 
change  in  temperature  does  not  cause  a  displacement  of  the 
equilibrium. 

The  reaction, 

C2H5OH  +  CHaCOOH  <=>  CH3COO.C2H5  +  H2O, 

is  accompanied  by  such  a  small  thermal  change  that  it  may  be 
considered  as  zero,  and  according  to  the  above  reasoning,  there 
should  be  only  a  very  slight  displacement  of  the  equilibrium  when 
the  temperature  is  varied.  Berthelot  found  that  at  10°  C., 
65.2  per  cent  of  the  alcohol  and  acid  are  changed  into  ester,  and 
at  220°  C.,  66.5  per  cent  of  the  mixture  is  transformed  into  ester. 
As  will  be  seen,  an  increase  of  210°  produces  hardly  any  displace- 
ment of  the  equilibrium, 


320  THEORETICAL  CHEMISTRY 

REFERENCES 

Chemical  Statics  and  Dynamics.  Mellor.  Chapters  IV  and  VIII. 
Thermodynamics  and  Chemistry.  Nernst.  Lectures  II  to  IV  incl. 
A  System  of  Physical  Chemistry.  Lewis.  Vol.  II.  Chapters  V  and  XII. 

PROBLEMS 

1.  When  2.94  mols  of  iodine  and  8.10  mols  of  hydrogen  are  heated  at 
constant  volume  at  444°  C.  until  equilibrium  is  established,  5^64  mols 
of  hydriodic  acid  are  formed.     If  we  start  with  5.30  mols  of  iodine  and 
7.94  mols  of  hydrogen,  how  much  hydriodic  acid  is  present  at  equilibrium 
at  the  same  temperature?  Ans.  9.49  mols. 

2.  At  2000°  C.,  and  under  atmospheric  pressure,  carbon  dioxide  is 
1.80  per  cent  dissociated  according  to  the  equation 

2  C02  <=*  2  CO  +  02. 

Calculate  the  equilibrium  constant  for  the  above  reaction  using  partial 
pressures.  Ans.  3  X  10~* 

3.  What  is  the  equilibrium  constant  in  the  preceding  problem,  if  the 
concentrations  are  expressed  in  mols  per  liter? 

4.  When  6.63  mols  of  amylene  and  1  mol  of  acetic  acid  are  mixed, 
0.838  mol  of  ester  is  formed  in  the  total  volume  of  894  liters.     How  much 
ester  will  be  formed  when  we  start  with  4.48"  mols  of  amylene  and  1  mol 
of  acetic  acid  in  the  volume  of  683  liters?  Ans.  0.8111  mol. 

5.  If  1  mol  of  acetic  acid  and  1  mol  of  ethyl  alcohol  are  mixed,  the 
reaction 

C2H5OH  +  CH3COOH  <=*  CH3COOC2HS  +  H20, 

proceeds  until  equilibrium  is  reached,  when  £  mol  of  ethyl  alcohol,  I  mol 
of  acetic  acid,  f  mol  of  ethyl  acetate,  and  f  mol  of  water  are  present.  If 
we  start  (a)  with  1  mol  of  acid  and  2  mols  of  alcohol;  (b)  with  1  mol  of 
acid,  1  mol  of  alcohol,  and  1  mol  of  water;  (c)  with  1  mol  of  ester  and  3 
mols  of  water,  how  much  ester  will  be  present  in  each  case  at  equilibrium? 

6.  In  the  reaction 

HCl  +  i  0  <dt  J  H20  +  Cl, 

we  find,  since  £  0  =  J  02,  for 

K= 


PHCI 


the  values  3.02  at  386°  C.  and  2.35  at  419°  C.    Calculate  the  heat  evolved 
by  the  reaction  under  constant  pressure.  Ans.  6896  cal. 

7.  Above  150°  C.  N02  begins  to  dissociate  according  to  the  equation 

N02  «*  NO  +  |  O*. 


HOMOGENEOUS  EQUILIBRIUM  321 

At  390°  C.  the  vapor  density  of  N02  is  19.57  (H  =  1),  and  at  490°  C. 
it  is  18.04.  Calculate  the  degree  of  dissociation  according  to  the  above 
equation  at  each  of  these  temperatures;  the  equilibrium  constants 
expressing  the  concentrations  in  mols  per  liter;  and  the  heat  of  dissocia- 
tion of  N02.  Ans.  ai  =  0.35,  «2  =  0.55,  K^  =  2.884  X  10-2. 

Kz  =  7.173  X  10~2.        Q  =  -9407  cal. 

8.  Water  vapor  and  carbon  monoxide  react  to  form  "  water-gas," 
according  to  the  equation, 

H2O  +  CO  =  H2  +  CO2. 

Calculate  the  composition  of  this  mixture  at  equilibrium,  at  a  temper- 
ature of  1000°  absolute,  and  under  a  total  pressure  (a)  of  0.1  atmosphere, 
and  (b)  of  10  atmospheres,  assuming  that  the  water  vapor  and  carbon 
monoxide  from  which  the  equilibrium  mixture  is  formed  were  originally 
present  in  equal  volumes. 
9  In  the  reaction 

2  S03  ^  2  S02  +  02, 

the  equilibrium  constant,  Kc,  has  been  found  to  have  the  following  values 
at  different  temperatures: 

Temp.  Kc 

627°  0.000316 

727  0.00354 

832  0.0280 

Calculate  the  corresponding  values  of  KP. 

10.  From  the  data  of  the  preceding  problem,  calculate  the  values  of 
Kc  and  Kv  at  500°,  and  also  calculate  the  heat  of  reaction  at  this  temper- 
ature. 


CHAPTER   XIII 
HETEROGENEOUS  EQUILIBRIUM 

Heterogeneous  Systems.  We  have  now  to  consider  equilibria 
in  systems  made  up  of  matter  in  different  states  of  aggregation. 
Such  systems  are  termed  heterogeneous  systems,  as  distinguished 
from  those  dealt  with  in  the  preceding  chapter,  where  the  compo- 
sition is  uniform  throughout.  The  physically  distinct  portions  of 
matter  involved  in  a  heterogeneous  system  are  known  as  phases, 
each  phase  being  homogeneous,  and  separated  from  the  other 
phases  by  definite  bounding  surfaces.  Thus,  ice,  liquid  water 
and  vapor,  constitute  a  physically  heterogeneous  system.  Another 
heterogeneous  system  is  formed  by  calcium  carbonate  and  its 
products  of  dissociation,  calcium  oxide  and  carbon  dioxide.  The 
equilibrium  between  a  solid,  its  saturated  solution,  and  vapor 
affords  an  illustration  of  a  still  more  complex  heterogeneous 
system. 

Application  of  the  Law  of  Mass  Action  to  Heterogeneous 
Equilibria.  It  has  been  shown  in  the  preceding  chapter,  that 
the  law  of  mass  action  may  be  applied  to  homogeneous  equilibria, 
provided  the  molecular  condition  of  the  reacting  substances  is 
known. 

When  we  attempt  to  apply  the  law  of  mass  action  to  hetero- 
geneous equilibria,  especially  where  solids  are  involved,  the 
problem  presents  difficulties.  In  his  investigation  of  the  dis- 
sociation of  calcium  carbonate,  according  to  the  equation, 

[CaC03]  <=±  [CaO]  +  (C02), 

Debray  *  showed,  that  just  as  every  liquid  has  a  definite  vapor 
pressure  corresponding  to  a  certain  temperature,  so  there  is  a 
definite  pressure  of  carbon  dioxide  over  calcium  carbonate,  at  a 
definite  temperature.  Furthermore,  the  pressure  was  found  to 
be  independent  of  the  amount  of  calcium  carbonate  present. 

*  Compt.  rend.,  64,  603  (1867). 
322 


HETEROGENEOUS  EQUILIBRIUM  323 

Guldberg  and  Waage*  showed,  that  the  law  of  mass  action  can 
be  applied  to  such  heterogeneous  equilibria,  provided  that  the 
active  masses  of  the  solids  present  are  considered  as  constant. 
Nernst  confirmed  this  statement  of  Guldberg  and  Waage  and 
showed  how  it  can  be  reconciled  with  experimental  facts.  In  a 
heterogeneous  equilibrium  involving  solids,  it  is  only  necessary  to 
consider  the  gaseous  phase,  since  the  active  mass  of  a  solid  is 
equivalent  to  its  concentration  in  the  gaseous  phase.  That  is, 
every  solid  is  to  be  looked  upon  as  possessing,  at  a  definite  tempera- 
ture, a  definite  vapor  pressure  which  is  entirely  independent  of  the 
amount  of  solid  present.  Such  substances  as  arsenic,  antimony, 
and  cadmium  are  known  to  have  appreciable  vapor  pressures  at 
relatively  low  temperatures,  and  it  is  quite  reasonable  to  sup- 
pose, that  every  solid  substance  exerts^^definita-^apor  pressure 
at  jt_de^nite-4£mperature,  even  though  we  have  no  method  suf- 
ficientlyj-efine^to  measura^udi  minute  pressures  . 

Since  the  active  mass  of  a  solid  remains  constant  so  long  as 
any  of  it  is  present,  the  application  of  the  law  of  mass  action 
to  certain  heterogeneous  equilibria  is,  in  general,  simpler  than 
its  application  to  homogeneous  systems.  The  truth  of  this  state- 
ment will  be  evident  after  a  few  typical  heterogeneous  systems 
have  been  considered. 

(a)  Dissociation  of  Calcium  Carbonate.     In  the  reaction, 

[CaCO31  ?=*  [CaO]  +  (CO2), 

let  TTi  and  irz  represent  the  pressures  due  to  the  vapor  of  calcium 
carbonate  and  calcium  oxide  respectively,  and  let  p  denote  the 
pressure  of  the  carbon  dioxide.  Applying  the  law  of  mass  action, 
we  obtain, 


But,  since  TTI  and  7r2  are  constant  at  any  one  temperature,  the 
equation  becomes, 

P  =   Kp', 

or,  the  equilibrium  constant,  at  any  one  temperature,  is  solely 
dependent  upon  the  pressure  of  the  carbon  dioxide  evolved.  The 
accompanying  table  gives  the  values  of  the  pressure  of  carbon 
dioxide  corresponding  to  various  temperatures. 

*  Loc.  cit. 


324 


THEORETICAL  CHEMISTRY 


DISSOCIATION   PRESSURES  OF  CaCO, 


Temperature, 
Degrees. 

Pressure  in 
Millimeters  of 
Mercury. 

547 

27 

610 

46 

625 

56 

740 

255 

745 

289 

810 

678 

812 

753 

865 

1333 

(b)  Dissociation  of  Ammonium  Hydrosulphide.  When  solid 
ammonium  hydrosulphide  is  heated,  it  is  almost  completely  dis- 
sociated into  ammonia  and  hydrogen  sulphide,  as  shown  by  the 
following  equation: 

[NH4HS]  <=>  (NH3)  +  (H2S). 

This  reaction  was  investigated  by  Isambert,*  who  found  that  the 
total  gas  pressure  at  25°.  1  C.,  is  equal  to  501  mm.  of  mercury. 
Since  the  partial  pressures  of  the  ammonia  and  hydrogen  sulphide 
are  necessarily  the  same,  each  must  be  approximately  equal  to 
250.5  mm.,  the  relatively  small  pressure  due  to  the  undissociated 
vapor  of  the  ammonium  hydrosulphide  being  neglected.  Let  IT 
be  the  partial  pressure  of  the  vapor  of  ammonium  hydrosulphide, 
and  let  pi  and  p%  be  the  partial  pressures  of  the  ammonia  and 
hydrogen  sulphide  respectively.  Applying  the  law  of  mass  action, 
we  have 


=  K,. 


(1) 


Since  TT  is  constant  at  any  one  temperature,  equation  (1)  becomes, 

Pi  •  P2  =  Kp'. 
According  to  Dalton's  law  of  partial  pressures,  we  have 

P    =    Pi   +  P2  +   7T, 

where  P  is  the  total  pressure.     Neglecting  the  relatively  small 
pressure  TT,  we  may  write, 

P  =  Pi  +  pi. 
*  Compt.  rend.,  93,  595,  730  (1881). 


HETEROGENEOUS  EQUILIBRIUM 
Hence,  since  p\  =  pzt 


325 


Substituting  these  values  in  equation  (1),  we  obtain 

=  62,750. 


(2) 


The  value  of  the  equilibrium  constant  may  be  checked,  by  observr 
ing  the  effect  on  the  system  of  the  addition  of  an  excess  of  either 
one  of  the  products  of  the  dissociation.  The  accompanying  table 
gives  the  results  of  a  few  of  Isambert's  experiments. 

DISSOCIATION  PRESSURES  OF  NH4HS 


Pressure  of 
Ammonia. 

Pressure  of 
Hydrogen 
Sulphide. 

PNHa.PH2S  =  Kp'. 

208 

294 

61,152 

138 

.  458 

63,204 

417 

146 

60,882 

453 

143 

64,779 

Mean  62,504 

As  will  be  seen,  the  mean  value  of  the  equilibrium  constant  agrees 
well  with  the  value  found  for  equivalent  amounts  of  the  products 
of  dissociation. 

(c)  Dissociation  of  Ammonium  Carbamate.  The  dissociation  of 
ammonium  carbamate  takes  place  according  to  the  equation, 

NH4 


NH 


This  dissociation  has  been  investigated  by  Horstmann.*  Apply- 
ing the  law  of  mass  action,  we  have 

Pi2 '  Pz  _  K  /o\ 

— ^ —  =  Kv>  w 

where  p\  and  p%  are  the  partial  pressures  of  ammonia  and  carbon 
dioxide  respectively,  and  where  TT  is  the  partial  pressure  of  ammo- 
nium carbamate.  Since  r  is  constant,  equation  (3)  becomes 

2/v>  V    f 

PI  p%  =    /Vp  . 
*  Lieb.  Ann.,  187,  48  (1877). 


326      .  THEORETICAL  CHEMISTRY 

If  P  denotes  the  total  gaseous  pressure,  and  TT  is  neglected  as  in 
the  preceding  example,  we  have,  since  three  mols  of  gas  are 
formed, 

*      4P2  P 

PI   =  — Q—     and     pz  =  -5-  • 

Substituting  these  values  in  equation  (3),  we  have 

4P* 
27  v  ' 

This  equation  has  also  been  tested  by  Isambert,*  by  adding  an 
excess  of  ammonia,  or  carbon  dioxide,  to  the  dissociating  system. 
He  found  that  the  value  of  the  equilibrium  constant  remains 
practically  constant.  Furthermore,  the  addition  of  a  foreign  gas 
was  shown  to  be  without  effect  on  the  dissociation. 

(d)  Dissociation  of  the  Hydrates  of  Copper  Sulphate.  Many 
interesting  examples  of  heterogeneous  equilibrium  are  furnished 
by  hydrated  salts.  Thus,  if  crystallized  copper  sulphate,  CuS04.- 
5  H2O,  be  placed  in  a  desiccator,  it  gradually  loses  water  of  crystal- 
lization, and  ultimately  the  anhydrous  salt  alone  remains.  If  a 
desiccator,  provided  with  a  manometer,  be  placed  in  a  thermo- 
stat, it  is  possible  to  observe  the  changes  in  vapor  pressure  ac- 
companying the  process  of  dehydration.  At  the  temperature  of 
50°  C.,  the  pressure  over  completely  hydrated  copper  sulphate  is 
found  to  remain  constant  at  47  mm.,  until  the  salt  has  been  de- 
prived of  two  molecules  of  water,  when  it  drops  abruptly  to  30 
mm.,  and  remains  constant  until  two  more  molecules  of  water 
have  been  lost.  It  then  drops  to  4.4  mm.,  and  remains  constant 
until  dehydration  is  complete. 

The  successive  stages  of  the  dehydration  are  shown  in  the  accom- 
panying diagram,  Fig.  83.  The  constant  pressures  observed  in 
the  dehydration  correspond  to  the  successive  equilibria  involved. 
At  50°  C.,  the  pentahydrate  and  the  trihydrate  are  in  equilibrium, 
a  pressure  of  47  mm.  being  maintained  so  long  as  any  of  the  penta- 
hydrate is  present.  When  all  of  the  pentahydrate  has  disap- 
peared, then  the  trihydrate  begins  to  undergo  dehydration  into  the 
monohydrate.  This  is  a  new  equilibrium,  and  the  pressure  of  the 
aqueous  vapor  necessarily  changes,  and  remains  constant  so  long 
as  any  trihydrate  remains.  The  last  stage  corresponds  to  the 

*  Loc.  cit. 


HETEROGENEOUS  EQUILIBRIUM 


327 


equilibrium  between  the  monohydrate  and  the  anhydrous  salt. 
The  following  equations  represent  the  three  successive  equilibria : 


(1)  CuSO4 

(2)  CuSO4 

(3)  CuS04 


5H2O 
3H2O 
H2O<= 


47  mm 


=±  CuSO4 
=±  CuSO4 
CuSO4 


3  H2O  +  2  H2O, 
H2O  +  2  H2O, 
H2O. 


30  mm 


4.5.  mm 


6H20 


3H2O 
Composition 

Fig.  -83 


1H20       OH2O 


Applying  the  law  of  mass  action  to  the  first  of  the  above  equi- 
libria, we  have 


in  which  in  and  7r2  denote  the  partial  pressures  due  to  the  hydrates, 
CuSO4 .5  H2O  and  CuSO4.3  H20  respectively,  and  p  denotes  the 
pressure  of  aqueous  vapor.  Since  TTI  and  ?r2  are  constant,  the 
above  expression  takes  the  form, 

p2  =  K,'. 

In  a  similar  manner,  it  may  be  shown  that  the  pressure  of  aqueous 
vapor  in  the  other  equilibria  must  be  constant.  It  must  be 
clearly  understood,  that  the  observed  pressure  is  only  definite 
and  fixed  when  two  hydrates  are  present.  If  the  dehydration 
were  conducted  at  another  temperature  than  50°  C.,  the  equilibrium 


328 


THEORETICAL  CHEMISTRY 


pressure  would  be  different.     The  vapor  pressure  curves  of  the 
different  hydrates  are  shown  diagrammatically  in  Fig.  84. 


Tee 


Temperature 
Fig.  84 

Heat  of  Dissociation  of  Solids.  When  the  products  of  the 
dissociation  of  a  solid  are  gaseous,  it  has  been  pointed  out  by  De 
Forcrand,*  that  the  ratio  of  the  heat  of  dissociation  of  1  mol  of 
solid  to  the  absolute  temperature  at  which  the  dissociation  pres- 
sure is  equal  to  1  atmosphere,  is  constant.  Or,  denoting  the  heat 
of  dissociation  by  Q,  and  the  absolute  temperature  by  T,  De  For- 
crand's  relation  may  be  expressed  thus, 


^  =  constant  =  33  - 


(5) 


Nernst  has  shown,  that  the  value  of  the  constant  in  this  relation 
is  not  independent  of  the  temperature.  Thus,  the  value  of  the 
ratio  at  100°  C.,  is  29.7,  while  at  1000°  C.,  it  is  37.7.  Up  to  the 
present  time,  no  expression  has  been  derived,  in  which  the  variation 
of  the  ratio  with  the  temperature  is  included. 


Ann.  Chim.  Phys.  [7],  28,  545. 


HETEROGENEOUS  EQUILIBRIUM 


329 


Distribution  of  a  Solute  between  Two  Immiscible  Solvents. 

When  an  aqueous  solution  of  succinic  acid  is  shaken  with  ether, 
the  acid  distributes  itself  between  the  ether  and  the  water  in  such 
a  way,  that  the  ratio  between  the  two  concentrations  is  always 
constant.  It  will  be  seen,  that  the  distribution  of  succinic 
acid  between  two  solvents  is  analogous  to  that_of_£u  substance- 
between  the  liquid  and  gaseous  phases  (see  page  154),  and  there- 
fore, the  laws  governing  the  latter  equilibrium  should  apply  equally 
to  the  former.  Nernst*  has  shown,  that  (a)  //  the  molecular 
weight  of  the  solute  is  the  same  in  both  solvents,  the  ratio  in  which  it 
distributes  itself  between  them  is  constant,  at  constant  temperature, 
or,  in  other  words,  Henryjs  law  is  applicable;  and  (b)  //  there  are 
several  solutes  in  solution,  the  distribution  of  each  solute  is  the  same  as 
if  it  were  present  alone.  This  is  clearly  Dalton's  law  of  partial 
pressures.  The  ratio  in  which  the  solute  distributes  itself  between 
the  two  solvents  is  termed  the  coefficient  of  distribution,  or  partition. 
The  following  table  gives  the  results  of  three  experiments  on  the 
distribution  of  succinic  acid  between  ether  and  water. 

DISTRIBUTION  OF  SUCCINIC  ACID 


Concentration 
in  Water. 

Concentration 
in  Ether. 

Distribution 
Coefficient. 

43.4 
43.8 
47.4 

7.1 
7.4 
7.9 

6.  n  n 

5.9, 
6.0 

As  will  be  seen,  the  distribution  coefficient  is  constant,  show- 
ing that  Henry's  law  applies.  When  the  molecular  weight  of  a 
solute  is  not  the  same  in  both  solvents,  the  distribution  coefficient 
is  not  constant,  and  conversely,  if  the  distribution  coefficient  is 
not  constant,  we  infer  that  the  molecular  weights  of  the  solute 
in  the  two  solvents  are  not  identical. 

Let  us  assume  that  a  solute  whose  normal  molecular  weight  is 
A,  when  shaken  with  two  immiscible  solvents,  undergoes  polymeri- 
zation in  one  of  them,  its  molecular  weight  being  An.  We  then 
have  the  equilibrium, 

An  <=±  nA : 


Zeit.  phys.  Chem.,  8,  110  (1891). 


330  THEORETICAL  CHEMISTRY 

applying  the  law  of  mass  action,  we  have 


CA 


(6) 


If  the  molecular  weight  in  one  solvent  is  twice  the  molecular 
weight  in  the  other,  then  n  =  2,  and 

Ci2  Cl 

—     or,      ~7=  =  constant. 

C2  VC2 

Thus,  Nernst  found  the  following  concentrations  of  benzoic  acid 
when  it  was  shaken  with  benzene  and  water. 

DISTRIBUTION  OF  BENZOIC  ACID 


ct  (Water). 

c2  (Benzene). 

£l. 

Cz 

Ci. 

V5' 

0.0150 
0.0195 
0.0289 

0.242 
0.412 
0.970 

0.062 
0.048 
0.030 

0.0305 
0.0304 
0.0293 

As  will  be  seen,  the  values  of  the  ratio  CI/QJ  steadily  decrease, 
while  on  the  other  hand,  the  values  of  the  ratio  Ci/\/C2  remain 
constant.  This  shows,  therefore,  that  benzoic  acid  has  twice  the 
normal  molecular  weight  in  benzene. 

The  Solution  of  a  Solid  in  a  Non-dissociating  Solvent.  When 
a  solid  is  introduced  into  a  non-dissociating  solvent,  it  con- 
tinues to  dissolve,  until  the  solution  becomes  saturated.  A 
condition  of  equilibrium  then  obtains,  and  the  rates  of  solution 
and  precipitation  are  the  same.  This  is  plainly  a  case  of  hetero- 
geneous equilibrium.  If  c  is  the  concentration  of  the  dissolved 
substance,  and  TT  is  the  concentration  of  the  undissolved  solid,  then 
according  to  the  law  of  mass  action, 


or,  since  IT  is  constant, 


c  =  Kc'. 


Variation  of  the  Constant  of  Heterogeneous  Equilibrium  with 
Temperature.    The  equation  of  the  reaction  isochore, 


d 


K) 


dT 


-Q 

RT2 


HETEROGENEOUS  EQUILIBRIUM  331 

which  expresses  the  variation  of  the  equilibrium  constant  with 
temperature,  applies  equally  well  to  heterogeneous  equilibria. 
The  following  examples  will  serve  to  illustrate  its  application  in 
such  cases. 

(a)  Dissociation  of  Ammonium  Hydrosulphide.     In  the  reaction 
representing  the  dissociation  of  ammonium  hydrosulphide, 

[NH4HS]  ^>  (NH3)  +  (H2S), 

let  pi  and  pz  be  the  partial  pressures  of  ammonia  and  hydrogen 
sulphide,  and  let  TT  be  the  partial  pressure  of  ammonium  hydro- 
sulphide.  Then  as  has  been  shown, 

K  '        ** 

Kp       :=T, 

where  P  is  the  total  gaseous  pressure.    From  the  following  data: 
Ti  =  273°  +    9°.5,         Pi  =  175  mm.  of  mercury, 

and 

T2  =  273°  +  25°.  1,         P2  =  501  mm.  of  mercury, 

we  have,  on  applying  the  reaction  isochore  equation,  and  solving 
fa  j 

4.581  [log  (^i)  -  log  ppj]  282.5  X  298.1 

=  298.1  -282.5 

or,  Qp  =  -  22,740  calories. 

This  result  agrees  well  with  the  value,  —  22,800  calories,  found 
by  direct  experiment. 

(b)  Solution  of  Succinic  Acid.     Concentration  and  temperature, 
are  the  factors  which  determine  the  equilibrium  in  a  saturated 
solution.     In  the  equation, 

-Q, 


dT         ~  RT2 

Kc  =  c,  where  c  is  the  concentration  of  succinic  acid  in  a  saturated 
solution.  The  following  experimental  data,  due  to  van't  Hoff, 
enables  us  to  calculate  the  heat  of  solution  of  the  acid. 

Ti  =  273°,  c  =  2.88  grams  per  100  grams  of  water, 
and 

T2  =  273°  +  8°.5,  c  =  4.22  grams  per  100  grams  of  water. 


332  THEORETICAL  CHEMISTRY 

Substituting  in  the  equation  of  the  reaction  isochore,  and  solving 
for  Qc,  we  have 

4.581  (log  4.22  -  log  2.88)  273  X  281.5 
"  ^°  =  281.5  -  273 

or 

Qv  =  -  6871  calories. 

The  value  of  the  heat  of  solution  for  1  mol  of  succinic  acid,  as 
found  by  direct  experiment,  is  —  6700  calories. 

The  Phase  Rule.  While  it  is  possible  to  apply  the  law  of 
mass  action  to  certain  heterogeneous  equilibria,  there  are  numerous 
cases  where  its  application  is  either  difficult,  or  impossible.  In 
dealing  with  such  heterogeneous  systems,  we  make  use  of  a  general- 
ization discovered  by  J.  Willard  Gibbs,*  late  professor  of  mathe- 
matical physics  in  Yale  University.  This  generalization  was  first 
stated  by  Gibbs  in  1874,  and  is  commonly  known  as  the  phase  rule. 
Before  entering  upon  a  discussion  of  the  phase  rule,  it  will  be 
necessary  to  define  a  few  of  the  terms  employed. 

The  composition  of  a  system  is  determined  by  the  number  of 
independent  variables,  or  components  involved.  Thus,  in  the 
system  —  ice,  water,  and  vapor  —  there  is  but  a  single  com- 
ponent. In  the  system, 

CaC03  +±  CaO  +  CO2, 

while  there  are  three  substances  which  constitute  the  equilibrium, 
only  two  of  these  need  be  considered  as  components,  for  the 
amount  of  any  one  constituent  is  not  independent  of  the  amounts 
of  the  other  two,  as  the  following  equations  show: 

CaO  +  CO2  =  CaCO3, 
CaC03  -  CaO  =  CO2, 
CaCO3  -  CO2  =  CaO. 

In  general,  the  components  are  chosen  from  the  smallest  number 
of  independently-variable  constituents  required  to  express  the 
composition  of  each  phase  entering  into  the  equilibrium,  even 
negative  quantities  of  the  components  being  permissible. 

The  number  of  variable  factors,  —  temperature,  pressure,  and 
concentration,  —  of  the  components  which  must  be  arbitrarily 
fixed  in  order  to  define  the  condition  of  the  system,  is  known  as 

*  Trans.  Connecticut  Academy,  Vols.  II  and  III,  1875-8. 


HETEROGENEOUS  EQUILIBRIUM  333 

the  degree  of  freedom  of  the  system.  For  example,  a  gas  has  two  de- 
grees of  freedom,  since  any  two  of  the  three  variables,  —  tempera- 
ture, pressure,  or  volume,  —  must  be  fixed  in  order  to  define  it;  a 
liquid  and  its  vapor  has  only  one  degree  of  freedom,  since  for 
equilibrium  at  a  certain  temperature,  there  can  be  but  a  single 
pressure;  while  in  a  system  consisting  of  a  substance  in  the  three 
states  of  aggregation,  equilibrium  can  exist  only  at  a  single  tem- 
perature and  pressure. 

Derivation  of  the  Phase  Rule.  The  following  derivation  of 
the  phase  rule  is  due  to  Nernst.*  Let  us  assume  a  complete  hetero- 
geneous equilibrium  made  up  of  y  phases  of  n  components,  and  let 
us  fix  our  attention  upon  any  one  of  the  y  phases.  This  phase  will 
contain  a  certain  amount -of  each  one  of  the  n  components,  the 
concentrations  of  which  may  be  designated  by  c\,  c%,  c3,  .  .  .  c». 
Since  we  have  assumed  complete  equilibrium  to  exist,  the  slightest 
change  in  concentration,  temperature,  or  pressure,  will  alter  the 
composition  of  this  phase.  This  may  be.  expressed  by  the 
equation, 

/  (ci,  cz,  Cr,  .  .  .  CH,  p,  T)  =  0, 

where  /  is  any  function  of  the  variables.  Since  any  change  ill 
one  phase,  implies  a  corresponding  change  in  the  remaining  y  —  1 
phases,  it  follows,  that  the  composition  of  all  the  phases  is  a  certain 
determined  function  of  the  same  variables.  The  above  equation 
is  then  of  the  form  ascribed  to  each  separate  phase,  and  since 
there  are  y  phases,  we  have  y  separate  equations.  There  are, 
however,  n  +  2  variables  in  each  equation,  so  that  if  y  =  n  +  2, 
that  is,  if  we  have  two  more  phases  than  components,  each  un- 
known quantity  has  a  definite  known  value.  In  this  case,  there 
is  only  one  value,  for  ci,  C2,  c3,  c4,  .  .  .  Cn,  p  and  T,  at  which  the 
system  can  be  in  equilibrium.  Hence,  when  n  components  are 
present  in  n  +  2  phases,  we  have  equilibrium  only  for  a  certain 
temperature,  a  certain  pressure,  and  a  certain  ratio  of  concentra- 
tions of  the  single  phases.  That  is,  n  +  2  phases  of  n  substances 
can  only  exist  at  a  certain  point  in  a  coordinate  system.  This 
point  is  termed  the  transition  point.  If  one  value  be  altered, 
then  one  phase  vanishes,  and  there  remain,  n  +  1  phases  of 
n  components,  whence  the  problem  becomes  indeterminate. 

*  Theoretical  Chemistry.  Nernst.  English  translation  by  Lehfeldt 
Page  611. 


334 


THEORETICAL  CHEMISTRY 


Thus  it  is  proved,  that  n  components  are  necessary  in  order  that 
a  system,  containing  n  +  1  phases,  may  exist  in  complete  equi- 
librium. 

The  phase  rule  may  be  stated  as  follows :  —  A  system  made  up 
of  n  components,  in  n  +  2  phases,  can  exist  only  when  pressure, 
temperature  and  concentration  have  definite  fixed  values;  a  system 
of  n  components ,'  in  n  +  1  phases,  can  exist  only  so  long  as  one  of  the 
factors  varies;  and  a  system  of  n  components,  in  n  phases,  can  exist 
only  so  long  as  two  of  the  factors  vary.  If  P  denotes  the  number  of 
phases,  C  the  number  of  components,  and  F  the  number  of  degrees 
of  freedom,  then  the  phase  rule  may  be  conveniently  summarized 
by  the  expression, 

C  -  P  +  2  =.F.  (7) 

Equilibrium  in  the  System,  Water,  Ice,  and  Vapor.  In  this 
system,  we  may  have  one,  two,  or  three  phases  present,  according 

.to  the  conditions. 
Under  ordinary  cir- 
cumstances of  temper- 
ature and  pressure, 
water  and  water  vapor 
are  in  equilibrium. 
The  vapor  pressure 
curve  of  water  is  rep- 
resented by  the  line 
OA,  in  the  pressure- 
temperature  diagram, 


Solid 


Vapor 


0.0075 
Temperature 

Fig.  85 


(Fig.  85).  It  is  only 
at  points  on  this  curve 
that  water  and  its 
vapor  are  in  equi- 
librium. Thus,  if  the 
pressure  be  reduced 
below  that  correspond- 


ing to  any  point  on  OA,  all  of  the  water  will  be  vaporized;  if,  on  the 
other  hand,  the  pressure  be  raised  above  the  curve,  all  of  the  vapor 
will  ultimately  condense  to  the  liquid  state.  When  the  temperature 
is  reduced  below  0°  C.,  only  ice  and  vapor  are  present,  the  curve 
OC,  representing  the  equilibrium  between  these  two  phases.  It  is 
to  be  observed  that  the  curve,  OC,  is  not  continuous  with  OA.  At 


HETEROGENEOUS  EQUILIBRIUM  335 

the  point  0,  where  the  two  curves  intersect,  ice,  water,  and  water 
vapor  are  in  equilibrium.  At  this  point,  ice  and  water  must  have 
the  same  vapor  pressure,  otherwise,  distillation  of  vapor,  from  the 
phase  having  the  higher  vapor  pressure  to  that  with  the  lower 
vapor  pressure,  would  occur,  and  eventually,  the  phase  having  the 
higher  vapor  pressure  would  disappear.  This  result  would  be 
in  contradiction  to  the  experimentally-determined  fact,  that  both 
solid  and  liquid  phases  are  in  equilibrium  at  the  point  0.  "The 
temperature  at  which  ice  and  water  are  in  equilibrium  with 
their  vapor,  under  atmospheric  pressure,  is  0°  C.  Since  increase 
of  pressure  lowers  the  freezing-point  of  water,  the  point  0,  repre- 
senting the  equilibrium  between  ice  and  water  under  the  pressure 
of  their  own  vapor,  viz.,  4.57  mm.,  must  be  a  little  above  0°  C. 
The  exact  temperature  corresponding  to  the  point  0  has  been 
found  to  be  0°.0075  C. 

The  change  in  the  melting-point  of  ice,  due  to  increasing  pressure, 
is  represented  by  the  line  OB.  This  line  is  inclined  toward  the 
vertical  axis  because  the  melting-point  of  ice  is  lowered  by  in- 
creased pressure.  The  point  0,  is  called  a  triple  point,  because 
there,  and  there  only,  three  phases  are  in  equilibrium.  As  is  well 
known,  water  does  not  always  freeze  exactly  at  0°  C.  If  the 
containing  vessel  is  perfectly  clean,  and  care  is  taken  to  exclude 
dust,  it  is  possible  to  supercool  water  several  degrees  below  its 
freezing-point,  and  measure  its  vapor  pressure. 

The  dotted  curve  OAf,  which  is  a  continuation  of  OA,  represents 
the  vapor  pressure  of  supercooled  water.  It  will  be  noticed,  that 
(1)  there  is  no  break  in  the  vapor-pressure  curve,  so  long  as  the 
solid  phase  does  not  separate,  and  (2)  the  vapor  pressure  of  super- 
cooled water,  which  is  an  unstable  phase,  is  greater  than  that  of 
ice,  the  stable  phase,  at  that  temperature. 

We  now  proceed  to  apply  the  phase  rule  to  this  system.  In  the 
formula,  C  -  P  +  2  =  F,  C  =  1.  It  is  evident  that  if  P  =  3, 
then  F  =  0;  or,  the  system  has  no  degree  of  freedom.  We  have 
seen  that  the  triple  point  0  represents  such  a  condition.  At  this 
point  ice,  water,  and  water  vapor  are  co-existent,  and  if  either 
one  of  the  variables,  temperature  or  pressure,  is  altered,  one  of 
the  phases  disappears;  in  other  words,  the  system  has  no  degree 
of  freedom.  Such  a  system  is  said  to  be  non-variant.  If,  in  the 
above  formula,  P  =  2,  then  F  =  1,  and  the  system  has  one  degree 
of  freedom,  or  is  univariant.  Any  point  on  any  one  of  the  curves, 


336 


THEORETICAL  CHEMISTRY 


OA,  OB,  or  OC,  represents  a  univariant  system.  Take,  for  ex- 
ample, a  point  on  the  curve  OA.  In  this  case,  the  temperature 
may  be  altered  without  altering  the  number  of  phases  in  equi- 
librium. If  the  temperature  is  raised,  a  corresponding  increase 
in  vapor  pressure  follows,  and  the  system  will  adjust  itself  to  some 
other  point  on  the  curve  OA.  In  like  manner,  the  pressure  may 
be  altered  without  causing  the  disappearance  of  one  of  the  phases. 
If,  however,  the  temperature  is  maintained  constant,  then  a  change 
in  the  pressure  will  cause  either  condensation  of  water  vapor,  or 
vaporization  of  liquid  water.  Under  these  conditions,  the  system 
has  only  one  degree  of  freedom.  Again,  if  P  =  1,  then  F  =  2, 
and  the  system  is  bivariant,  or  has  two  degrees  of  freedom.  The 
areas  included  between  the  curves  in  the  diagram  are  examples  of 
bivariant  systems.  Let  us  consider  the  vapor  phase.  The  tempera- 
ture may  be  fixed  at  any  desired  value  within  the  vapor  area  AOC, 
and  the  pressure  may  be  altered  along  a  line  parallel  to  the  vertical 
axis  without  causing  a  change  in  the  number  of  phases,  provided 


12000 


10000 


8000 


6000 


4000 


2000 


-40          -30          -20  -10  0  10 

Temperature 
Fig.  86 


20 


40 


the  curves  OA  and  OC  are  not  intersected.  The  lower  limit  of 
the  sublimation  curve  OC,  is  theoretically  determined  by  the 
absolute  zero,  while  the  vaporization  curve  OA,  terminates  at 
the  critical  temperature,  364.3°,  corresponding  to  a  pressure  of 
194.6  atmospheres. 


HETEROGENEOUS  EQUILIBRIUM 


337 


Investigations  conducted  by  Tammann,*  and  Bridgman,t  with 
a  view  to  determining  the  upper  limit  of  the  fusion  curve  OB,  have 
revealed  the  existence  of  at  least  five  different  crystalline  modi- 
fications of  ice,  all  of  which,  with  the  exception  of  ordinary  ice, 
are  denser  than  water.  These  five  forms  of  ice  have  been  desig- 
nated Ice  I  (ordinary  ice),  Ice  II,  Ice  III,  Ice  V,  and  Ice  VI. 
Although  Tammann' s  experiments  indicate  the  possibility  of  the 
existence  of  Ice  IV,  it  has  not  yet  been  satisfactorily  established. 
The  accompanying  diagram,  Fig.  86,  shows  the  relationships  which 
these  different  modifications  of  ice  bear  to  each  other.  The 
coordinates  of  the  important  points  are  given  below. 


Phases 

Point 

Temp. 

Press. 

Ice  I,  liquid  and  vapor  

o 

+  0.0075 

4.579  mm.  Hg. 

Ice  I,  liquid,  Ice  III  

c 

-  22° 

2115  kg.  per  cm.2 

Ice  III,  liquid,  Ice  V  

D 

-  17° 

3530 

Ice  V,  liquid,  Ice  VI  

E 

+    0.16° 

6380           " 

Ice  I,  Ice  II,  Ice  III  

F 

-  34.7° 

2170           " 

Ice  II,  Ice  III,  Ice  V 

G 

—  24.3° 

3510 

The  System,  Sulphur  (Rhombic,  Monoclinic),  Liquid  and 
Vapor.  This  system  is  more  complicated  than  the  preceding 
one-component  system,  since  there  are  two  solid  phases,  in  addition 
to  the  liquid  and  vapor  phases.  At  ordinary  temperatures,  rhom- 
bic sulphur  is  the  stable  modification.  When  this  is  heated 
rapidly  it  melts  at  115°  C.,  but  if  it  is  maintained  in  the  neighbor- 
hood of  100°  C.,  it  gradually  changes  into  monoclinic  sulphur, 
which  melts  at  120°  C.  Monoclinic  sulphur  can  be  kept  indefi- 
nitely at  100°  C.  without  undergoing  change  into  the  rhombic 
modification,  or  in  other  words,  it  is  the  stable  phase  at  this  temper- 
ature. It  is  evident,  therefore,  that  there  must  be  a  temperature 
above  which  monoclinic  sulphur  is  the  stable  form,  and  below  which 
rhombic  sulphur  is  the  stable  modification.  This  temperature 
at  which  both  rhombic  and  monoclinic  modifications  are  in  equi- 
librium with  each  other  and  with  their  vapor,4  is  termed  the 
transition  point.  Its  value  has  been  determined  to  be  95°.6  C. 
The  change  from  one  form  into  the  other  is  relatively  slow,  so 
that  it  is  possible  to  measure  the  vapor  pressure  of  rhombic  sul- 

.    Ann.  der  Physik,  (4)  2,  1424  (1900). 
Proc.  Am.  Acad.,  47,  411  (1912). 


338  THEORETICAL  CHEMISTRY 

phur  up  to  its  melting-point,  and  that  of  monoclinic  sulphur 
below  its  transition  point.  The  vapor  pressure  of  solid  sulphur, 
although  very  small,  has  been  measured  as  low  as  50°  C. 

The    complete   pressure-temperature    diagram   for   sulphur   is 
shown  in  Fig.  87.     At  the  point  0,  rhombic  and  monoclinic  sul- 


131? 


Rhombic  Sulphur 


Vapor 


Temperature 
Fig.  87 

phur  are  in  equilibrium  with  sulphur  vapor,  this  being  a  triple 
point,  analogous  to  the  point  0,  in  Fig.  86.  The  vapor  pressure 
curves  of  rhombic  and  monoclinic  sulphur  are  represented  by  OB 
and  OA,  respectively.  The  dotted  curve  OAf,  which  is  a  continu- 
ation of  OA,  is  the  vapor-pressure  curve  of  monoclinic  sulphur  in 
a  metastable  region.  In  like  manner,  OB'  represents  the  vapor- 
pressure  curve  of  rhombic  sulphur  in  the  metastable  condition,  Ef 
being  a  metastable  melting-point.  As  in  the  pressure-temper- 
ature diagram  for  water,  the  metastable  phases  have  the  higher 


HETEROGENEOUS  EQUILIBRIUM  339 

vapor  pressure.  The  effect  of  increasing  pressure  on  the  transi- 
tion point  0,  is  represented  by  the  line  OC.  This  is  termed  a 
transition  curve.  Since  increase  in  pressure  raises  the  transition 
point,  the  line  OC,  slopes  away  from  the  vertical  axis.  The 
effect  of  increase  of  pressure  on  the  melting-point  of  mono- 
clinic  sulphur,  is  shown  by  the  curve  AC.  This  also  slopes  away 
from  the  vertical  axis,  but  the  change  in  the  melting-point  of 
monoclinic  sulphur  produced  by  a  given  change  in  pressure 
being  less  than  the  corresponding  change  in  the  transition  point, 
the  two  curves,  OC  and  AC,  intersect  at  the  point  C.  The 
point  C  corresponds  to  a  temperature  of  131°  C.,  and  a  pres- 
sure of  400  atmospheres.  The  vapor-pressure  curve  of  stable 
liquid  sulphur  is  represented  by  the  curve  AD.  The  vapor- 
pressure  curve  of  the  metastable  liquid  phase  is  represented  by 
the  curve  A Bf,  which  is  continuous  with  AD.  The  diagram  is 
completed  by  the  curve  B'C,  which  represents  the  effect  of  pressure 
on  the  metastable  melting-point  of  rhombic  sulphur.  Mono- 
clinic  sulphur  does  not  exist  above  the  point  C.  Hence,  when 
liquid  sulphur  is  allowed  to  solidify  at  pressures  exceeding  400  at- 
mospheres, the  rhombic  modification  is  formed,  whereas  under 
ordinary  pressures  the  monoclinic  modification  appears  first. 

The  phase  rule  enables  us  to  state  the  exact  conditions  required 
for  equilibrium,  and  to  check  the  results  of  experiment.  Thus, 
according  to  the  formula,  C  —  P  +  2  =  F,  since  C  =  1,  the  system 
will  be  non-variant  when  P  =  3.  Since  there  are  four  phases  in- 
volved, theoretically  any  three  of  these  may  be  co-existent  and 
four  triple  points  are  possible.  The  theoretically-possible  triple 
points  are  as  follows : 

(1)  Rhombic  sulphur,  monoclinic  sulphur,  and  vapor  (0); 

(2)  Rhombic  sulphur,  monoclinic  sulphur,   and  liquid   (C); 

(3)  Rhombic  sulphur,  liquid,  and  vapor  (#')>' 

(4)  Monoclinic  sulphur,  liquid  and  vapor  (A). 

In  this  particular  system  all  of  the  four  possible  triple  points  can 
be  realized  experimentally.  That  this  is  the  case,  is  due  to  the 
comparative  slowness  of  the  change  from  rhombic  to  monoclinic 
sulphur  above  the  triple  point.  If  this  change  were  rapid,  it  is 
evident  that  all  of  the  theoretically-possible  non-variant  systems 
could  not  be  realized  experimentally. 

As  in  the  case  of  water,  the  curves  in  the  diagram  represent 


340  THEORETICAL  CHEMISTRY 

univariant  systems  and  the  areas,  bivariant  systems.  The  student 
is  advised  to  tabulate  the  univariant  and  bivariant  systems  repre- 
sented in  the  pressure-temperature  diagram  for  sulphur. 

Two-Component  Systems.  When  we  apply  the  phase  rule  to 
systems  of  two  components,  we  obtain,  2  —  P  +  2  =  F,  or,  in  other 
words,  we  find  that  the  maximum  number  of  degrees  of  freedom 
possible,  is  three.  Therefore,  in  order  to  represent  a  two-com- 
ponent system  completely,  three  rectangular  coordinate  axes, 
representing  temperature,  pressure  and  composition,  will  be  re- 
quired. While  three-dimensional  models  constructed  in  this 
manner  undoubtedly  give  the  most  complete  representation  of 
two-component  systems,  it  has  been  found  that  the  various 
phase  relations  can  be  depicted  clearly  and  satisfactorily  by 
passing  planes  through  such  space  models,  perpendicular  to  the 
three  coordinate  axes.  It  follows,  therefore,  that  there  will  be 
three  different  sets  of  such  diagrams  corresponding  to  each  model; 
namely,  the  P-T  diagrams,  the  T-C  diagrams  and  the  P-C 
diagrams.  Of  these  diagrams,  the  P-T  and  T-C  diagrams  are 
generally  the  most  important.  Among  the  different  types  of 
two-component  systems  which  have  been  studied,  may  be  men- 
tioned systems  involving  anhydrous  salts  and  water,  hydrated 
salts  and  water,  volatile  solutes,  two  liquid  phases,  consolute 
liquids,  and  solid  solutions. 

We  shall  confine  ourselves  to  the  consideration  of  two-com- 
ponent systems,  consisting  of  non-volatile  solid  solutes  and  li- 
quid solvents.  These  systems  may  be  conveniently  classified,  as 
follows : 

Class  I.  The  two  components  crystallize  together  as  a  mixture 
of  the  pure  components. 

Class  II.  The  two  components  crystallize  together  forming 
definite  chemical  compounds. 

Class  III.  The  two  components  crystallize  together  forming 
mixed  crystals. 

Class  I.  —  KC1  +  H20.  The  P-T  diagram  of  this  system  is 
shown  in  Fig.  88,  together  with  the  similar  diagram  for  water 
alone,  the  latter  being  represented  by  dotted  lines.  At  the  triple 
point,  0,  ice  and  water  have  the  same  vapor  pressure.  Similarly, 
a  solution  at  its  freezing-point  has  the  same  vapor  pressure  as 
the  ice  which  separates.  The  intersection  of  the  vapor-pressure 
curve  of  ice,  OB,  and  the  vapor-pressure  curve  of  the  solution 


HETEROGENEOUS  EQUILIBRIUM 


341 


of  the  anhydrous  salt,  0"A",  determines  a  new  triple  point 
0".  Since  the  presence  of  the  dissolved  salt  tends  to  diminish 
the  vapor  pressure  of  water,  the  curve  0"  A"  is  situated  below 
the  curve  OA,  and  for  the  same  reason,  the  triple  point  0"  is 
found  to  the  left  of  0.  If  an  excess  of  dissolved  substance  is 
present,  all  of  the  liquid  phases  which  are  formed  will  of  necessity 
be  saturated  solutions.  When  these  solutions  finally  freeze,  they 


T 


Temperature 
Fig.  88 


will  furnish,  not  pure  ice,  but  a  mixture  of  ice  and  solid  salt,  known 
as  a  cryohydrate.  By  a  partial  freezing,  we  can  obtain  the  system : 
Solid  salt,  ice,  saturated  solution  and  vapor,  or  in  other  words, 
a  system  of  n  +  2  phases,  which  can  exist  only  at  the  freez- 
ing temperature  of  the  saturated  solution,  T' ',  and  at  the  pres- 
sure p't  corresponding  to  the  vapor  pressure  of  ice  and  that  of 
the  saturated  solution.  These  conditions  are  represented  by  the 
quadruple  point  0' '.  If  we  pass  from  the  point  0',  increasing  the 
temperature  and  pressure  as  prescribed  by  the  curve  O'A',  the 
ice  will  disappear,  while  the  salt,  the  saturated  solution,  and 
the  vapor  will  furnish  a  series  of  three-phase  systems.  Again  start- 
ing from  the  point  0',  and  lowering  the  temperature  and  the  pres- 
sure, as  indicated  by  the  curve  O'B,  the  liquid  phase  will  disappear, 
while  the  solid  salt,  ice,  and  vapor  will  constitute  another  series 


342  THEORETICAL  CHEMISTRY 

of  three-phase  systems.  This,  of  course,  is  on  the  supposition 
that  the  vapor  pressure  of  the  solid  salt  is  negligible.  Finally,  a 
considerable  increase  in  pressure  will  cause  a  slight  lowering  of 
the  temperature  of  the  quadruple  point,  0',  as  shown  by  the 
curve  0'C". 

All  possible  non-saturated  solutions  of  the  salt  will  be  repre- 
sented by  points  within  the  area,  AOO'A'.  Thus,  let  0"  A" 
represent  the  vapor-pressure  curve  of  a  dilute  aqueous  solution  of 
the  salt.  The  freezing-point  of  this  solution  is  represented  by 
the  point  0",  while  0"C"  represents  the  variation  of  the  freezing- 
point  of  the  solution  with  pressure.  The  following  table  sum- 
marizes the  possibilities  indicated  by  the  phase  rule : 

(a)  Areas.     (N  -  P  +  2  =  F),  2-2  +  2  =  2,  i.e.,  divariant 
systems : 

Salt- vapor,  below  A'O'B  and  above  T-axis, 
Solution- vapor,  between  AOA'O', 
Solution-ice,  between  COO'C', 
Ice-salt,  between  C'O'B  and  P-axis. 

(b)  Curves.     (N  -  P  +  2  =  F),  2-3  +  2  =  1,  i.e.,  mono- 
variant  systems: 

Solution- vapor-salt,  A'O', 
Solution-ice-salt,  C'O', 
Ice-salt-vapor,  BO', 
Solution-ice-vapor,  00'. 

(c)  Points.      (N  -  P  +  2  =  F),   2-4  +  2  =  0,    i.e.,   non- 
variant  system,  0'. 

The  P-T  diagram  (Fig.  88)  having  been  discussed,  we  now 
turn  to  the  T-C  diagram  for  the  same  system,  Fig.  89.  In  this 
diagram,  the  abscissae  represent  concentrations,  and  the  ordinates, 
temperatures.  For  convenience,  corresponding  points  in  Figs. 
89  and  90  will  be  designated  by  the  same  letters.  The  equi- 
librium between  ice,  water  and  water  vapor,  is  represented  by 
the  point  0.  As  the  proportion  of  salt  is  increased,  the  tempera- 
ture of  equilibrium  is  lowered  along  the  curve  00'.  Ultimately, 
a  point  will  be  reached  at  which  the  solution  becomes  saturated, 
and  on  further  addition  of  salt  it  will  no  longer  dissolve  but 
will  remain  in  contact  with  the  ice  and  saturated  solution.  This 
is  the  cryohydric  point,  and  represents  the  lowest  temperature 
which  can  be  obtained  in  this  particular  system.  The  diagram 


HETEROGENEOUS  EQUILIBRIUM 


343 


-20 


is  completed  by  the  solubility  curve  of  the  salt,  O'A'.  Each 
point  on  this  curve  represents  the  concentration  of  a  saturated 
solution  down  to  the  cryohydric  temperature. 

The  meaning  of  the  concentration-temperature  diagram  may 
be  made  clearer  by  a  consideration  of  the  behavior  of  a  solution 
when  gradually  cooled.  Let ' 
a  represent  a  dilute  solution 
of  the  anhydrous  salt.  On 
lowering  the  temperature  80 
along  ab,  no  change  will 
occur  until  the  curve  00 '  GO 
is  reached;  then,  ice  will  g 
begin  to  separate,  and  as  1 4Q 
the  cooling  is  continued,  the  | 
composition  of  the  solution  3 
will  change  along  00 '  until  20 
it  reaches  the  cryohydric 
point  0' '.  Here,  both  salt  ° 
and  ice  will  separate,  and 
the  solution  will  solidify 
completely  at  the  tempera- 
ture corresponding  to  the 
point  0'.  In  like  manner, 
if  we  start  with  a  concentrated  solution,  represented  by  the 
point  c,  and  cool  along  cd,  no  change  will  take  place  until 
the  curve  O'A'  is  reached;  then  solid  salt  will  separate  and 
the  composition  of  the  solution  will  alter  along  O'A',  until  the 
temperature  is  reduced  to  that  corresponding  to  the  cryohydric 
point,  when  the  whole  solution  will  solidify  as  in  the  previous 
case. 

This  phenomenon  was  first  systematically  investigated  by  Guth- 
rie,*  who  concluded  that  such  mixtures  having  constant  com- 
position and  definite  melting-point,  are  chemical  compounds. 
Therefore,  he  proposed  to  call  them  cryohydrates.  It  has  since 
been  shown,  that  cryohydrates  are  not  definite  chemical  com- 
pounds. Among  the  various  reasons  which  have  been  advanced 
to  prove  the  incorrectness  of  Guthrie's  views,  the  following  are 
the  most  convincing.  —  (1)  the  physical  properties  of  a  cryohy- 
drate  are  the  mean  of  the  corresponding  properties  of  the  con- 
*  Phil.  Mag.  [4],  49,  ]  (1875);  [5],  i,  49  and  2,  211  (1876). 


20 


40  60 

Percentage  KC1 

Fig.   89 


100 


344 


THEORETICAL  CHEMISTRY 


stituents,  this  being  rarely  true  of  chemical  compounds;  (2)  the 
lack  of  homogeneity  of  a  cryohydrate  can  be  detected  under  the 
microscope;  and  (3)  the  constituents  are  seldom  present  in  simple 
molecular  proportions. 

Class  II.  —  FeCl3  +  H20.     An  interesting  example  of  this  class 
of  two-component  systems   is   furnished   by  mixtures   of  ferric 

chloride  and  water. 
This  system  has  been 
very  carefully  investi- 
gated by  Roozeboom.* 
The  concentration- 
temperature  diagram, 
plotted  from  Rooze- 
boom's  data,  is  given 
in  Fig.  90.  The  freez- 
ing-point of  pure 
water  is  represented 
by  A,  and  the  lower- 
ing of  the  freezing- 
point  produced  by 

"2°r *        '  the  addition  of  ferric 

chloride    is    indicated 

-40 h    \  /  by    the    curve    AB. 

At  the  cryohydric 
temperature,  —  55°  C., 
ice,  Fe2Cl6  •  12  H2O, 
saturated  solution, 
and  vapor  are  in 
equilibrium,  and  the  system  is  non-variant.  On  adding  more 
ferric  chloride,  the  ice  phase  disappears,  and  the  univariant  sys- 
tem, Fe2Cl6  •  12  H2O,  saturated  solution,  and  vapor  results.  The 
equilibrium  is  represented  by  the  curve  BC,  which  may  be  re- 
garded as  the  solubility  curve  of  the  dodecahydrate.  On  con- 
tinuing the  addition  of  ferric  chloride,  the  temperature  continues 
to  rise,  until  the  point  C  is  reached.  Here  the  composition  of  the 
solution  is  identical  with  that  of  the  dodecahydrate,  and,  there- 
fore, the  temperature  corresponding  to  this  point,  37°  C.,  may  be 
looked  upon  as  the  melting-point  of  Fe2Cl6  •  12  H2O.  Further 


10  15  ^2Q 

Mols  Fe2Cl6to  100  mols  H2O 

Fig.   90 


25 


Zeit.  phys.  Chem.,  4,  31  (1889);   10,  477  (1892). 


HETEROGENEOUS  EQUILIBRIUM  345 

addition  of  ferric  chloride,  will  naturally  lower  the  melting-point, 
and  the  equilibrium  will  alter  along  the  curve  CD.  It  is  thus 
possible  to  have  two  saturated  solutions,  one  of  which  contains 
more  water,  and  the  other  less,  than  the  hydrate  which  is  in  equi- 
librium with  the  solution.  These  solutions  are  both  stable 
throughout,  and  are  nowhere  supersaturated.  Roozeboom  was 
the  first  investigator  to  discover  a  saturated  solution  containing 
less  water  than  the  solid  hydrate  with  which  it  is  in  equilibrium. 
This  discovery  led  him  to  define  supersaturation  as  follows:  — 
"  A  solution  is  supersaturated  with  respect  to  a  solid  phase,  at  a 
given  temperature,  if  its  composition  is  between  that  of  the  solid 
phase  and  the  saturated  solution. "  At  the  point  D,  the  curve 
reaches  another  minimum,  which  is  analogous  to  the  point  5, 
except  that  the  heptahydrate,  Fe2Cl6  •  7  H2O,  takes  the  place 
of  ice.  Here  we  have  equilibrium  between,  the  dodecahydrate, 
the  heptahydrate,  saturated  solution,  and  vapor,  and  the  system 
is  non-variant.  On  further  addition  of  ferric  chloride,  another 
maximum  is  reached  at  E,  corresponding  to  the  melting-point  of 
the  heptahydrate.  In  a  similar  manner,  two  other  maxima  at 
greater  concentrations  of  ferric  chloride,  reveal  the  existence  of 
the  hydrates,  FeCl6  •  5  H2O,  and  Fe2Cl6  •  4  H2O.  At  the  three 
remaining  quadruple  points  the  following  phases  are  in  equi- 
librium: At  F,  Fe2Cl6  •  7  H2O,  Fe2Cl6  •  5  H2O,  saturated  solution 
and  vapor;  at  H,  Fe2Cl6  •  5  H2O,  Fe2Cl6  •  4  H2O,  saturated  solution 
and  vapor;  and  at  K,  Fe2Cl6  •  4  H2O,  Fe2Cle,  saturated  solution 
and  vapor.  The  solubility  of  the  anhydrous  salt  is  represented 
by  the  curve,  KL.  Metastable  solubility  and  melting-point 
curves  are  represented  by  dotted  lines. 

The  student  should  apply  the  phase  rule  to  this  system.  If  a 
fairly  dilute  solution  of  ferric  chloride  is  evaporated  at  31°  C.,  the 
water  gradually  disappears,  and  a  residue  of  the  dodecahydrate 
remains.  This  residue  then  liquefies,  and  again  dries  down,  the 
composition  of  the  residue  corresponding  to  the  heptahydrate: 
on  further  standing,  the  phenomenon  is  repeated,  the  final  and 
permanent  residue  having  a  composition  corresponding  to  the 
pentahydrate.  The  dotted  line,  ab,  shows  the  isothermal  along 
which  the  composition  varies.  It  would  have  been  a  difficult 
matter  to  explain  the  alternations  of  moisture  and  dryness,  ob- 
served in  this  experiment,  without  the  concentration-temperature 
diagram. 


346 


THEORETICAL  CHEMISTRY 


Class  III.  In  all  of  the  two-component  systems  thus  far  con- 
sidered, the  composition  of  the  solid  phase  has  remained  constant, 
while  that  of  the  liquid  phase  has  undergone  continuous  variation. 
In  the  systems  belonging  to  Class  III,  however,  the  solid  phase 
which  separates  from  the  solution,  on  cooling,  is  found  to  contain 
both  components  in  continuously  varying  proportions.  It  fol- 
lows, therefore,  that  two-component  systems  belonging  to  this 
class  consist  of  two  solutions,  one  a  liquid  solution  and  the  other  a 
solid  solution,  in  equilibrium  with  each  other.  Obviously,  the 
complete  temperature-concentration  diagram  must  contain  two 
curves,  one  of  which  corresponds  to  the  liquid,  and  the  other  to 
the  solid,  solution.  The  upper  curve,  corresponding  to  the  li- 
quid solution,  is  commonly  called  the  liquidus  curve,  while  the 
lower  curve,  corresponding  to  the  solid  solution,  is  known  as  the 
solidus  curve.  The  substitution  of  the  terms,  crystallization- 
point  curve,  and  melting-point  curve,  for  the  terms  liquidus  and 
solidus  curves,  is  greatly  to  be  preferred.  The  region  above  the 
curves,  represents  liquid  solutions,  while  the  region  below  the 
curves,  represents  solid  solutions.  There  are  three  distinct  types 
of  melting-point  curves  for  solid  solutions  in  which  the  two  com- 
ponents are  miscible  in  all  proportions:  namely,  Type  I,  in  which 

the  freezing-points  of  all 
mixtures  lie  between  those 
of  the  two  components; 
Type  II,  in  which  a  certain 
mixture  has  a  maximum 
freezing-point;  and  Type 
III,  in  which  a  particular 
mixture  has  a  minimum 
freezing-point. 

The  general  character- 
istics of  a  two-component 
system  belonging  to  Type 
I,  may  be  illustrated  by  the 
temperature  -  concentration 
diagram  of  the  system 
AgCl  —  NaCl,  as  shown  in  Fig.  91.  If  the  temperature  of  a 
fused  mixture  of  these  two  salts,  having  the  composition  rep- 
resented by  the  point  I  on  the  crystallization  point  curve, 
be  gradually  lowered,  a  solid  phase,  having  the  composition 


900 


800 


700 


600 


400 


NaCl 


AgCl 


40  60 

Percentage  NaCl 

Fig.  91 


100 


HETEROGENEOUS  EQUILIBRIUM 


347 


20  40  60  80 

Percentage  d-  Carvoxime 
Fig.    92 


100 


represented  by  s,  will  separate  out.  If  the  temperature  be 
lowered  very  slowly,  so  as  to  insure  the  homogeneity  of  the 
solid  phase,  the  composi-  95 
tion  of  the  final  por- 
tion of  the  solution  to 
solidify,  will  correspond 
to  that  of  the  point  s', 
and  the  composition  of 
the  remaining  liquid  solu- 
tion will  be  represented 
by  the  point  V.  The 
temperature  interval  rep- 
resented by  the  line,  Is', 
is  known  as  the  crystal- 
lization interval.  If  care 
is  not  taken  to  cool  the 
solution  gradually,  the 
solid  phase  will  not  become  homogeneous,  owing  to  the  slow  rate 
of  diffusion  of  the  two  components  in  the  mixture,  and  conse- 
quently the  final  tem- 
perature of  solidification 
will  be  considerably  lower 
than  that  corresponding 
to  the  point  s'.  This 
precaution  is  of  great 
practical  importance  in 
connection  with  the  prep- 
aration of  alloys,  where 
the  rate  of  diffusion  of 
the  constituents  is  neces- 
sarily slow. 

Illustrations     of     the 
two  remaining  types  of 
binary    mixtures    which 
form  mixed  crystals  are  shown  in  the  accompanying  diagrams. 
The  diagram  of  the  system,  d-  and  Z-carvoxime  (Ci0Hi4N.OH), 
shown  in  Fig.  92,*  furnishes  an  example  of  Type  II,  while  Type 
III  is  illustrated  by  the  system  HgBr2  —  HgI2,  t  a  diagram  of 
*  Adrian!,  Zeit.  phys.  Chem.,  33,  469  (1900). 
f  Reinders,  ibid.  32,  494  (1900). 


260 


250 


[240 


210 


20  40  60  80 

Molecular  percentage  HgI2 

Fig.  93 


100 


348 


THEORETICAL  CHEMISTRY 


240 


200 


160 


100 


which  is  shown  in  Fig.  93.  It  will  be  seen,  that  the  curves  shown 
in  Figs.  91,  92  and  93,  are  exactly  analogous  to  the  three  types  of 
boiling-point  curves  of  binary  mixtures  which  have  been  discussed 
in  an  earlier  chapter,  (see  p.  161). 

A  large  number  of  binary  systems  are  known  in  which  the 
two  components  do  not  form  a  continuous  series  of  mixed  crystals. 
This  case,  which  is  perfectly  analogous  to  that  of  the  partial  mis- 

cibility  of  liquids,  may 
be  illustrated  by  the 
system  KNO3  -  T1NO3, 
a  diagram  of  which  is 
shown  in  Fig.  94.  At 
the  temperature  corre- 
sponding to  C,  two  solid 
solutions  having  composi- 
tions corresponding  to,  D 
and  E,  separate  out,  the 
temperature  remaining 
constant  until  all  of  the 
liquid  has  solidified.  On 
lowering  the  temperature 
below  C,  the  two  solid 
solutions,  which  were  in  equilibrium  at  C,  undergo  a  change  in 
composition,  as  indicated  by  the  dotted  lines. 

Three-Component  Systems.  According  to  the  phase  rule,  the 
number  of  degrees  of  freedom  in  a  three-component  system  will 
be  given  by  the  expression,  3  —  P  +  2  =  F.  It  is  evident, 
therefore,  that  the  system  will  become  invariant  when  five 
phases,  —  the  maximum  number  possible  in  a  three-component 
system  —  are  co-existent.  Since,  at  least  one  phase  must  be 
present,  the  maximum  number  of  degrees  of  freedom  will  be  four, 
the  variables  being  temperature,  pressure,  and  the  ratios  of  two  of 
the  components  to  the  remaining  component  in  any  single  phase. 
In  the  graphic  representation  of  three-component  systems,  use 
is  commonly  made  of  the  triangular  diagram.  This  consists  of  an 
equilateral  triangle,  each  side  of  which  is  divided  into  one  hundred 
equal  parts,  corresponding  to  100  per  cent  of  each  of  the  three 
pure  components,  which  are  represented  by  the  apices  of  the 
triangle.  For  example,  if  it  is  desired  to  represent  on  a  triangular 
diagram,  a  ternary  mixture  consisting  of  x  per  cent  of  A,  y  per 


20  40  60  80 

Molecular  percentage  T1NO  3 
Fig.  94 


HETEROGENEOUS   EQUILIBRIUM 


349 


Fig.   95 


cent  of  B  and  z  per  cent  of  C,  we  would  first  measure  off  on  A  B,  in 
the  triangular  diagram  shown  in  Fig.  95,  x  units  from  B  and  y 
units  from  A,  and  then  through  the  points  so  obtained,  draw 
lines  parallel  to  BC  and  AC  respectively.  The  point,  P,  at  which 
these  two  lines  intersect,  will  rep- 
resent the  ternary  mixture  hav- 
ing the  composition  given  above. 
Should  it  be  desired  to  represent, 
simultaneously,  the  variation  of 
the  equilibrium  with  temperature, 
this  can  readily  be  accomplished 
by  plotting  the  latter  on  an  axis 
perpendicular  to  the  plane  of  the 
triangular  diagram. 

Owing    to    the    complexity    of 
three-component  systems,  we  shall 
limit  our  treatment  of  these  systems  to  a  brief  consideration  of 
one  or  two  relatively  simple  cases. 

The    system,    chloroform-water-acetic    acid,    which    has    been 
thoroughly   investigated    by  Wright,*    furnishes   an    interesting 

example  of  a  ternary 
liquid  mixture,  in 
which  the  three  com- 
ponents form  a  single 
pair  of  partially  mis- 
cible  liquids.  At  or- 
dinary temperatures, 
water  and  chloroform 
are  only  slightly  sol- 
uble in  each  other, 
while  both  of  these 
70  so  90  l  B  H2°  liquids  are  miscible 
with  acetic  acid  in 
all  proportions.  The 
equilibrium  relations  in  this  system,  at  18°  and  under  con- 
stant pressure,  are  shown  graphically  in  Fig.  96.  Since  both 
temperature  and  pressure  are  assumed  to  be  constant,  there 
are  three  degrees  of  freedom.  If  we  have  but  a  single  homo- 
geneous phase,  it  will  ha.ve  two  degrees  of  freedom,  and  in 
*  Proc.  Roy.  Soc.,  49,  174  (1891);  50,  375  (1892). 


CHC13 


Art     10     20      30     40     50     6 

Fig.  96 


350  THEORETICAL  CHEMISTRY 

consequence,  will  be  represented  by  some  point  within  a  definite 
region.  In  like  manner,  if  we  have  two  liquid  phases,  the  system 
will  be  univariant,  and  therefore,  will  be  represented  by  some  point 
on  the  curve  aKb.  On  adding  water  to  chloroform,  we  obtain 
a  homogeneous  solution,  provided  the  amount  of  water  does  not 
exceed  that  corresponding  to  complete  saturation,  represented 
by  the  point  a.  If  further  amounts  of  water  are  added,  the 
mixture  will  separate  into  two  layers,  one  of  which  consists  of  a 
saturated  solution  of  water  in  chloroform,  and  the  other,  a 
saturated  solution  of  chloroform  in  water.  The  composition  of 
these  two  conjugate  solutions  is  represented  by  the  points,  a  and  b. 
If  now,  acetic  acid  is  added,  it  will  distribute  itself  between  the 
two  layers,  forming  two  conjugate  ternary  solutions  of  chloroform, 
water  and  acetic  acid  which  are  in  equilibrium  with  each  other. 
These  solutions  are  represented  by  two  points,  such  as  a'  and  bf, 
within  the  triangular  diagram.  The  line  joining  a'  and  b'  is  known 
as  a  tie-line;  as  is  generally  the  case,  this  line  is  not  parallel  to 
the  base  of  the  triangle  ABC.  As  more  and  more  acid  is  added, 
the  tie-lines  become  shorter  and  shorter,  and  ultimately,  when 
the  compositions  of  the  two  conjugate  solutions  become  identical, 
they  shrink  to  the  single  point,  K.  The  point  K  is,  therefore,  a 
critical  point,  since  further  addition  of  acetic  acid  will  result  in 
the  formation  of  a  single  homogeneous  phase.  Any  point  within 
the  curve,  represents  a  ternary  mixture  which  will  separate  into 
two  conjugate  liquid  phases,  while  any  point  without  the  curve, 
represents  a  single  homogeneous  liquid  phase.  Since  we  have  a 
bivariant  system  made  up  of  three  components,  existing  in  two 
liquid  phases  and  one  vapor  phase,  it  follows,  that  the  composi- 
tion of  the  two  conjugate  solutions,  and  the  total  vapor  pressure, 
will  depend,  not  only  on  the  temperature,  as  in  the  case  of  two- 
component  systems,  but  also  on  the  initial  composition  of  the 
mixture.  If  the  total  vapor  pressure  be  assumed  to  remain  con- 
stant, the  composition  of  the  different  conjugate  ternary  solu- 
tions will  vary  with  the  temperature  alone.  At  the  point  K, 
where  the  two  liquid  phases  become  identical,  the  number  of 
degrees  of  freedom  is  reduced  to  one,  and  therefore,  the  critical 
point  is  seen  to  be  dependent  upon  the  temperature  alone.  As 
has  been  pointed  out,  a  ternary  mixture  whose  composition  is 
represented  by  the  point  P,  will  separate  into  two  conjugate 
ternary  solutions  having  the  compositions,  a'  and  b'.  If  a  line  be 


HETEROGENEOUS  EQUILIBRIUM  351 

drawn  from  A  through  P,  cutting  the  isotherm  aKb  in  E  and  F, 
and  the  line  BC  in  G,  it  can  be  shown,  that  in  all  mixtures  repre- 
sented by  points  on  AG,  the  ratio  of  the  components,  B  and  C,  is 
constant,  and  equal  to  GB/GC.  If  to  a  mixture  of  water  and 
acetic  acid  whose  composition  is  represented  by  the  point  G}  in- 
creasing amounts  of  chloroform  be  added,  we  shall  obtain  in  suc- 
cession, (1)  a  series  of  homogeneous  solutions  between  G  and  F}  (2) 
a  series  of  heterogeneous  mixtures  of  two  conjugate  solutions 
between  F  and  E,  and  (3)  another  series  of  homogeneous  solu- 
tions between  E  and  A. 

Owing  to  the  fact  that  acetic  acid  does  not  distribute  itself 
equally  between  water  and  chloroform,  the  critical  point  K  lies 
below  the  highest  point  of  the  curve  Hence,  by  selecting  a 
homogeneous  solution  in  which  the  concentration  of  the  acetic 
acid  is  greater  than  that  corresponding  to  the  critical  point  K, 
but  less  than  that  represented  by  the  summit  of  the  curve,  it  is 
possible,  by  adding  the  proper  amounts  of  the  three  components, 
to  maintain  the  concentration  of  the  acetic  acid  constant,  and  yet 
pass  along  a  horizontal  line  from  a  homogeneous  region,  through 
a  heterogeneous  region,  into  a  homogeneous  region  again.  This 
phenomenon  is  known  as  retrograde  solubility,  and  is  encountered 
in  all  cases  where  the  critical  point  lies  below  the  highest  point 
of  the  isothermal  curve. 

We  will  now  consider  very  briefly,  two  other  typical  three-com- 
ponent systems,  involving  two,  and  three  pairs  of  partially  miscible 
liquids,  respectively.  Water 

The  system  water-aniline- 
phenol,  which  has  been  invest- 
igated by  Schreinemakers,* 
may  be  taken  as  an  example 
of  a  three-component  system 
containing  two  pairs  of  par- 
tially miscible  liquids,  namely 
water  and  aniline,  and  water 
and  phenol.  This  system  is 
represented,  diagrammatically,  in  Fig.  97,  three  different  iso- 
thermals  being  given.  The  interpretation  of  this  diagram  should 
be  fully  written  out  by  the  student. 

The  system,  water-ether-succinic  nitrile,  also  investigated  by 
*  Zeit.  phys.  Chem..  29,  577  (1898). 


352  THEORETICAL  CHEMISTRY 

Schreinemakers,*  furnishes  an  example  of  a  three-component 
system  containing  three  pairs  of  partially  miscible  liquids.  This 
system  is  represented  in  the  diagram,  shown  in  Fig.  98.  The 
compositions  of  all  mixtures  forming  single  homogeneous  solutions 
are  represented  by  points  in  the  unshaded  areas  of  the  triangular 
diagram.  All  points  lying  within  the  shaded  quadrilateral  areas, 
represent  mixtures  which  form  two  liquid  layers,  while  all  points 

included    within    the     shaded 

Nitrile  ,    . 

triangular  area,  represent  mix- 
tures which  break  up  into  three 
liquid  layers  of  the  composition 
a,  b  and  c. 

Alloys.     The  phase  rule  has 
proven    to    be    of    inestimable 
value  in    the    investigation   of 
Fi     98  intermetallic     mixtures,     com- 

monly   known    as    alloys.     In 

the  study  of  these  substances,  the  determination  of  the  freez- 
ing-point curve  is  of  fundamental  importance,  since,  from  its 
characteristics,  an  insight  into  the  nature  of  the  solid  and  liquid 
phases  involved  can  generally  be  obtained.  It  has  already  been 
pointed  out,  that  the  freezing-point  curves  of  two-component 
systems  may  conveniently  be  divided  into  the  three  following 
types:  —  Type  I,  in  which  the  pure  components  separate  out 
from  the  mixture;  Type  II,  in  which  the  components  form  one 
or  more  compounds  with  each  other;  and  Type  III,  in  which  the 
components  form  mixed  crystals.  Representatives  of  each  of 
these  types  are  found  among  the  alloys. 

Owing  to  the  fact,  that  the  temperatures  at  which  most  alloys 
melt  are  relatively  high,  and  also  because  of  the  difficulty  of  bring- 
ing about  the  separation  of  the  different  solid  phases  free  from 
contamination  with  the  other  phases,  it  is  customary  to  employ 
the  method  of  thermal  analysis  in  the  investigation  of  the  freezing- 
points  of  alloys.  This  method  is  based  upon  the  experimental 
determination  of  the  cooling  curves  of  molten  mixtures  of  the 
given  metals  in  varying  proportions.  The  curve  obtained  by 
plotting  temperature  against  time,  when  a  two-component  li- 
quid system  is  allowed  to  cool  slowly  by  radiation,  is  called  the 
cooling  curve.  This  curve  is  logarithmic  when  the  temperature 
*  Zeit.  phys.  Chem.,  25,  543  (1808). 


HETEROGENEOUS  EQUILIBRIUM 


353 


of  the  environment  is  constant,  but  becomes  rectilinear,  if  the 
surrounding  temperature  is  regularly  and  progressively  lowered. 
So  long  as  the  system  consists  of  a  single  liquid  phase,  the  cooling 
curve  will  remain  smooth  and  continuous,  as  shown  by  AB  in 
Fig.  99a,  but  should  a  second  phase  appear,  a  sudden  disconti- 
nuity in  the  curve  will  occur  simultaneously,  as  shown  by  BC  in 
the  diagram.  If  the  new  phase  has  the  same  composition  as  the 
liquid  phase,  the  temperature  will 
remain  constant,  as  in  the  dia- 
gram, until  all  of  the  liquid  phase 
has  disappeared.  On  the  other 
hand,  if  the  new  phase  differs  in 
composition  from  that  of  the 
liquid  phase,  the  cooling  curve 
will  undergo  a  somewhat  abrupt 
change  in  direction,  due  to  the 
fact,  that  the  composition  of  the 
liquid  phase  alters  continuously 
as  the  temperature  falls.  If  a 
third  phase  is  formed,  the  tempera- 
ture of  the  system  will  remain 
constant,  until  one  of  the  phases 
has  completely  disappeared.  The 

cooling  curve  of  a  two-component  liquid  system, 'in  which  the  first 
phase  to  separate  has  a  different  composition  from  that  of  the 
liquid  phase,  is  shown  in  Fig.  99b.  Here  EF  represents  the  cooling 
curve  of  the  liquid,  F,  the  appearance  of  the  second  phase  differ- 
ing in  composition  from  that  of  the  liquid  phase,  while  H  repre- 
sents the  appearance  of  a  third  phase.  During  the  separation  of 
the  latter,  the  temperature  of  the]system  remains  constant,  after 
which  it  cools  uniformly  along  IK. 

The  System,  Zinc-Cadmium.  A  brief  consideration  of  the 
two-component  system,  zinc-cadmium,  will  serve  to  illustrate  the 
application  of  the  method  of  thermal  analysis  to  the  study  of 
alloys.  In  this  system,  the  two  components  are  not  miscible  in 
the  solid  state  and  do  not  form  intermetallic  compounds.  In 
order  to  determine  the  curves  of  equilibrium,  mixtures  of  the  two 
metals,  in  varying  proportions,  are  fused,  and  then  allowed  to  cool 
slowly,  the  rate  of  cooling  being  measured  by  means  of  a  thermo- 
couple, one  junction  of  which  is  maintained  at  constant  tempera- 


Time 
Fig.  99 


354 


THEORETICAL   CHEMISTRY 


ture,  while  the  other  junction  is  placed  in  the  mixture.  The  ter- 
minals of  the  thermocouple  are  connected  to  a  sensitive  galva- 
nometer, graduated  to  read  directly  in  degrees,  and  the  rate  of 
cooling  is  followed  by  the  movement  of  the  needle  of  the  galva- 
nometer. The  cooling  curves,  thus  obtained,  for  each  of  the  two 
components,  and  for  nine  mixtures  containing  increasing  amounts 
of  cadmium,  are  plotted  in  the  diagram,  shown  in  Fig.  100.  Let 

Zn 


7          8 


9       Cd 


us  take,  for  example,  cooling  curve  No.  2,  corresponding  to  a  liquid 
mixture  which  is  relatively  rich  in  zinc.  As  the  temperature  falls, 
a  point  will  ultimately  be  reached  at  which  pure  zinc  begins  to 
crystallize,  and  since  the  temperature  remains  constant  during  the 
crystallization,  a  break  occurs  in  the  cooling  curve.  This  first 
break  is  the  freezing-point  of  the  mixture,  and  determines  a  point 
on  the  curve  AO.  As  zinc  continues  to  separate,  the  composition 
of  the  mixture  changes  along  AO,  until  0  is  reached:  here  the 
mixture  is  saturated  with  respect  to  cadmium,  and  both  metals 
separate  as  a  conglomerate,  having  the  same  composition  as  the 
fused  mixture.  The  separation  of  the  latter  phase  causes  the 
second  break  in  the  cooling  curve,  the  temperature  remaining 
constant  until  the  entire  mass  has  solidified.  It  will  be  noticed, 


HETEROGENEOUS  EQUILIBRIUM 


355 


that  this  system  is  the  exact  analogue  of  the  system  —  anhydrous 
salt  and  water.  The  point  0,  which  corresponds  to  the  cryo- 
hydric  point,  is  called  the  eutectic  point,  and  the  composition  of 
the  mixture  having  this  minimum  melting-point,  is  known  as  the 
eutectic  mixture.  From  the  cooling  curve  diagram  we  can  con- 
struct the  equilibrium  diagram,  shown  in  Fig.  101. 


500 

450 

«400 

I350 

^300 
250 


Melt 


+  Cd     - 


460 

400 

350 

B 

300 

250 


10 


20 


30 


40          50          60 
Percentage  Cd 

Fig.  101 


70 


80 


90 


100 


The  System,  Gold-Aluminium.  A  binary  system  belonging  to 
Type  II,  in  which  the  components  form  several  compounds  with 
each  other,  is  the  system,  gold-aluminium,  which  has  been  studied 
by  Heycock  and  Neville.*  The  concentration-temperature  dia- 
gram of  this  system,  which  is  shown  in  Fig.  102,  reveals  the 
existence  of  definite  compounds,  Au5Al2,  Au2Al,  and  AuAl2,  cor- 
responding to  the  points  D,  E  and  H,  respectively.  The  discon- 
tinuities at  B  and  G  suggest  the  possibility  of  two  other  com- 
pounds, viz.,  Au4Al  and  AuAl.  The  diagram  shows  that  the 
following  substances  will  crystallize  in  succession  from  the  molten 
alloy,  these  being  the  different  solids  with  which  the  liquid  mixture 
is  saturated  in  its  successive  stages  of  equilibrium: 

Curve  AB,  pure  gold  at  A; 
Curve  BC,  Au4Al,  nearly  pure  at  B; 
Curve  CD,  Au5Al2  or  AusAl3,  nearly  pure  at  D; 
*  Phil.  Trans.  A.,  194,  201  (1900). 


356 


THEORETICAL  CHEMISTRY 


Curve  DEF,  Au2Al,  pure  at  E; 

Curve  FG,  AuAl,  maximum  undetermined, 

Curve  GHI,  AuAl2,  pure  at  H] 

Curve  7J,  Al,  pure  at  J. 


Au 


Composition 
Fig.   102 


Al 


The  points  C,  F,  and  /  represent  non-variant  systems,  the  melt- 
ing-points of  the  respective  eutectic  alloys  being  527°,  569°,  and 
647°.  This  system  in  many  respects  resembles  the  system  — 
ferric  chloride  and  water. 

For  a  further  discussion  of  the  phase  rule  and  its  applications, 
the  student  must  consult  the  references  listed  below. 


REFERENCES 

The  Phase  Rule  and  its  Applications.     Findlay. 

The  Phase  Rule.     Bancroft. 

Physical  Chemistry  of  the  Metals.     Schenck. 

ters  II  and  III. 
Metallic  Alloys.     Gulliver. 

Metallography.     Desch.     Chapters  II  to  V  inclusive. 
The  Crystallization  of  Iron  and  Steel.     Mellor. 


Translated  by  Dean.     Chap- 


HETEROGENEOUS  EQUILIBRIUM 


357 


PROBLEMS 

1.  The  vapor  pressure  of  solid  NH4HS  at  25°.  1  is  50.1  cm.  Assuming 
that  the  vapor  is  practically  completely  dissociated  into  NH3  and  H2S, 
calculate  the  total  pressure  at  equilibrium  when  solid  NH4HS  is  allowed 
to  dissociate  at  25°.  1  in  a  vessel  containing  ammonia  at  a  pressure  of 
32  cm.  Ans.  59.5  cm. 

'2.  In  the  partition  of  acetic  acid  between  CC14  and  water,  the  con- 
centration of  the  acetic  acid  in  the  CC14  layer  was  c  gram-molecules  per 
liter  and  in  the  corresponding  water  layer  w  gram-molecules  per  liter. 


0.292 

4.87 


0.363 
5.42 


0.725 

7.98 


1.07 
9.69 


1.41 
10.7 


Acetic  acid  has  its  normal  molecular  weight  in  aqueous  solutions.  From 
these  figures  show  that,  at  these  concentrations,  the  acetic  acid  in  the 
carbon  tetrachloride  solution  exists  as  double  molecules. 


100 


788.4° 


Mj?  (633  ) 


/ 


20  40  60  80  100 

Atomic  percentage  Mg 

Fig.  103 


Acetic  acid  distributes  itself  between  water  and  benzene  in  such  a 
manner  that  in  a  definite  volume  of  water  there  are  0.245  and  0.314  gram 
of  the  acid,  while  in  an  equal  volume  of  benzene  there  are  0.043  and  0.071 
gram.  What  is  the  molecular  weight  of  acetic  acid  in  benzene,  assuming 
it  to  be  normal  in  water?  Ans.  121.3. 

4.  The  salt  Na2HP04.12  H20  has  a  vapor  pressure  of  15°  of  8.84  mm., 
and  at  17°. 3  of  10/53  mm.  Calculate  the  heat  of  vaporization,  i.e.,  the 
thermal  change  during  the  loss  of  1  mol  of  water  of  crystallization  by 
evaporation.  Ans.  -  12,651  cal. 


358 


THEORETICAL  CHEMISTRY 


5.  The  solubility  of  boric  acid  in  water  is  38.45  grams  per  liter  at  13°, 
and  49.09  grams  per  liter  at  20°.  Calculate  the  heat  of  solution  of  boric 
acid  per  mol.  Am.  —  5822  cal. 


1300 


1200 


Mn  (1260°) 


Cu  (1084°) 


(870°) 


10          20          30          40          50          60          70 
Atomic  percentage  Mn 

Fig.  104 


90         100 


-50 


Cd  (322°: 


Hg(-38  ) 


20  40  60 

Atomic  percentage  of  Cd 

Fig.    105 


80 


100 


6.  Plot  the  pressure-temperature  diagram  for  calcium  carbonate  from 
the  table  given  on  p.  324,  and  apply  the  phase  rule. 

7.  Is  it  possible  to  decide  by  the  phase  rule  whether  the  eutectic  alloy 
is  a  mixture  or  a  compound? 


HETEROGENEOUS  EQUILIBRIUM  359 

8.  By  means  of  the  principle  of  Le  Chatelier,  verify  as  far  as  possible 
the  statement  made  on  p.  337  relative  to  the  densities  of  the  different 
forms  of  ice  discovered  by  Tammann  and  Bridgman.     Make  use  of  the 
diagram  shown  in  Fig.  336. 

9.  Which  has  the  greater  density,  (a)  monoclinic  sulphur  or  rhombic 
sulphur;    (b)  rhombic  sulphur  or  liquid  sulphur;    (c)  monoclinic  sulphur 
or  liquid  sulphur? 

10.  How  do  you  explain  the  action  of  freezing  mixtures,  such  as  ice 
and  common  salt? 

11.  Why  is  it  that  cryohydrates  can  be  used  as  constant  temperature 
baths? 

12.  Interpret  the  temperature-composition  diagram  for  the  system, 
tin-magnesium,  shown  in  Fig.  103. 

13.  Interpret  the  temperature-composition  diagram  for  the  system, 
copper-manganese,  shown  in  Fig.  104. 

14.  Interpret  the  temperature-composition  diagram  for  the  system, 
mercury-cadmium,  shown  in  Fig.  105. 


CHAPTER  XIV 
CHEMICAL  KINETICS 

Velocity  of  Reaction.  In  the  two  preceding  chapters  we  have 
considered  the  equilibrium  which  is  established  when  the  speeds 
of  the  direct  and  reverse  reactions  have  become  equal.  We  now 
proceed  to  consider  the  velocity  of  individual  reactions.  By  far 
the  greater  number  of  the  reactions  between  inorganic  substances, 
proceed  with  such  rapidity  that  it  is  impossible  to  measure  their 
velocities.  Thus,  when  an  acid  is  neutralized  by  a  base,  the  indi- 
cator changes  color  almost  instantly.  There  are  a  few  well- 
known  reactions  which  are  exceptions  to  this  rule;  among  these 
may  be  mentioned  the  oxidation  of  sulphur  dioxide,  and  the  de- 
composition of  hydrogen  peroxide.  Both  of  these  reactions  are 
well  adapted  to  kinetic  experiments.  In  organic  chemistry,  on 
the  other  hand,  slow  reactions  are  the  rule  rather  than  the  excep- 
tion. Thus  the  reaction  between  an  alcohol  and  an  acid,  forming 
an  ester  and  water,  proceeds  very  slowly  under  ordinary  condi- 
tions, and  the  progress  of  the  reaction  may  be  easily  followed. 
By  means  of  the  law  of  mass  action,  it  is  possible  to  derive  equa- 
tions expressing  the  velocity  of  a  reaction  at  any  moment,  in  terms 
of  the  concentrations  of  the  reacting  substances  present  at  that  time. 

Let  the  equation, 

Ai  +  A8^Ai'  +  A*', 

represent  a  reversible  reaction  and  let  a,  6,  c,  and  d  be  the  respec- 
tive initial  concentrations  of  the  reacting  substances  A\,  Az,  A\, 
and  A2'.  The  velocity  of  the  direct  reaction  will  then  be 

g  =  k  (a  -  x)  (b  -  x),  (1) 

where  k  is  the  velocity  constant,  and  dx  is  the  infinitely  small 
increase  in  the  amount  of  x,  during  the  infinitely  small  interval 
of  time  dt.  Similarly,  the  velocity  of  the  reverse  reaction  will  be 

far1 

^  =  fc,  (c  +  x)  (d  +  x)  (2) 

360 


CHEMICAL  KINETICS  361 

It  is  evident,  that  the  substances  on  the  right-hand  side  of  the 
equation  will  exert  an  ever-increasing  influence  upon  the  velocity 
of  the  direct  reaction,  which  must  accordingly  decrease.  When, 
however,  the  velocities  of  the  direct  and  reverse  reactions  become 
equal,  equilibrium  will  be  established,  and  the  ratio  of  the  amounts 
of  the  reacting  substances  on  the  two  sides  of  the  equation  will 
remain  constant.  The  total  velocity  due  to  these  opposing  reac- 
tions will  be 

d-^  =  ^-d^=k(a-x)(b-x)-k1(c  +  x)(d  +  x),    (3) 

7  -y 

and  at  equilibrium,  when  -5-  =  0, 

k  (a  -  x)  (b  -  x)  =  k,  (c  +  x)  (d  +  x), 
or 

(c  +  x)  (d  +  x)  _k_r  /Ax 

(a  -  x)  (b  -  x)  ~  k,  ~ 

This  equation  has  been  thoroughly  tested  in  the  two  preceding 
chapters.  Thus,  in  the  reaction, 

C2H5OH  +  CH3COOH  <=±  CH3COOC2H5  +  H20, 

Ke  has  been  shown  to  have  the  value,  2.84,  at  ordinary  temper- 
atures. The  velocity  constants  of  the  direct  and  reverse  reac- 
tions have  also  been  determined,  the  values  being,  k  =  0.000238 
and  ki  =  0.0000815.  When  these  values  are  substituted  in  the 

k 

equation,  •=-  =  Kc,  we  obtain  Kc  =  2.92,  a  value  which  agrees 
KI 

well  with  that  found  by  direct  experiment.  The  application  of 
equation  (3)  is  much  simplified  by  the  fact,  that  most  reactions 
proceed  nearly  to  completion  in  one  direction,  so  that  the  term, 
ki(c  +  x)  (d  +  x),  will  be  so  small  that  it  may  be  neglected.  We 
then  have, 

g  =  k  (a  -  x)  (b  -  x),  (5) 

an  equation  expressing  the  velocity  of  the  direct  reaction  in  terms 
of  the  concentrations  of  the  reacting  substances. 

Unimolecular  Reactions.  The  simplest  type  of  chemical 
reaction  is  that  in  which  only  one  substance  undergoes  change, 
and  in  which  the  velocity  of  the  reverse  reaction  is  negligible. 


362 


THEORETICAL  CHEMISTRY 


The  decomposition  of  hydrogen  peroxide  is  an  example  of  such  a 
reaction.  In  the  presence  of  a  catalyst,  such  as  certain  unor- 
ganized ferments,  or  colloidal  platinum,  hydrogen  peroxide  de- 
composes as  represented  by  the  equation,  u  -r  ^^^  ,- 

H2O2  <^  H2O  +  i  02. 

This  reaction  is  usually  allowed  to  take  place  in  dilute  aqueous 
solution,  so  that  there  is  no  appreciable  alteration  in  the  amount 
•of  solvent  throughout  the  entire  course  of  the  reaction.  Further- 
more, the  activity  of  the  catalyst  remains  constant,  so  that  the 
course  of  the  reaction  is  wholly  determined  by  the  concentration 
of  the  hydrogen  peroxide.  A  very  satisfactory  catalyst  is  catal- 
ase,  an  enzyme  derived  from  blood.  The  concentration  of  hydro- 
gen peroxide  present  at  any  time  during  the  reaction,  can  be  deter- 
mined very  simply  by  removing  a  definite  portion  of  the  reaction 
mixture,  adding  an  excess  of  sulphuric  acid  to  destroy  the  activity 
of  the  catalase,  and  then  titrating  with  a  standard  solution  of 
potassium  permanganate. 
The  following  table  gives  the  results  of  such  an  experiment:  — 

DECOMPOSITION  OF  H202  BY  CATALASE 


t  (minutes). 

cc.  KMn04. 

cc.  KMnO4. 

k 

0 

46.1 

0 

5 

37.1 

9.0 

0.0435 

10 

29.8 

16.3 

0.0438 

20 

19.6 

26.5 

0.0429 

30 

12.3 

33.8 

0.0440 

50 

5.0 

41.1 

0.0444 

Mean  0.0437 

The  second  column  of  the  table  gives  the  number  of  cubic 
centimeters  of  the  potassium  permanganate  solution  required  to 
oxidize  25  cc.  of  the  reaction  mixture,  when  the  time  intervals 
recorded  in  the  first  column  have  elapsed  after  the  introduction 
of  the  catalyst.  Since  the  numbers  in  the  second  column,  repre- 
sent the  actual  concentration  of  hydrogen  peroxide  present  at 
the  end  of  the  successive  intervals  of  time,  it  is  evident,  that  the 
difference  between  these  numbers  and  46.1  cc.  —  the  initial 
concentration  of  hydrogen  peroxide  / —  will  give  the  amounts  of 
peroxide  decomposed  in  those  intervals.  These  numbers  are 


CHEMICAL  KINETICS  363 

recorded  in  the  third  column  of  the  table.  It  will  be  seen,  that 
as  the  concentration  of  the  hydrogen  peroxide  decreases,  the  rate 
of  the  reaction  diminishes.  Thus,  in  the  first  interval  of  10  min- 
utes, an  amount  of  hydrogen  peroxide  corresponding  to  46.1  - 
29.8  =  16.3  cc.  of  potassium  permanganate  is  decomposed,  while 
in  the  second  interval  of  10  minutes,  the  amount  of  hydrogen 
peroxide  decomposed  is  equivalent  to  29.8  —  19.6  =  10.2  cc.  of 
potassium  permanganate.  Since  only  a  single  substance  is  under- 
going change,  equation  (5)  simplifies  to  the  following  form: 

dx      7  /  \ 

^  =  k  (a  -  x). 

It  is  impossible  to  apply  the  equation  in  this  form,  since  in  order 
to  obtain  accurate  titrations,  dt  must  be  taken  fairly  large,  and 
during  this  interval  of  time,  a  —  x  will  have  diminished.  Ap- 
proximate values  of  k  may  be  obtained,  by  taking  the  average 
value  of  a  —  x,  during  the  interval  dt,  within  which  an  amount  dx 
of  hydrogen  peroxide  is  being  decomposed.  For  example,  let 
us  take  the  interval  between  5  and  10  minutes;  dx  =  16.3  — 
9.0  =  7.3  cc.,  dt  —  5  min.,  and  the  average  value  of  a  —  x  is 

37.1  +  29.8 

2 =  33.45  cc. 

Substituting  in  the  equation, 

dx      ,  ,          , 
jj  =  k  (a  -  x), 

we  have 

^  =  k  X  33.45, 
o 

and 

k  =  0.0436. 

Similarly,  taking  the  next  interval  between  10  and  20  minutes; 
dx  =  26.5  -  16.3  =  10.2  cc.,  dt  =  10  minutes,  and  the  average 

rtQ   O       I        1  Q   f\ 

value  of  a  —  x  is,  -  -  =24.7  cc.     Substituting  in  the 

equation  as  before,  we  obtain  | 


364  THEORETICAL  CHEMISTRY 

and 

k  =  0.0413. 

As  will  be  seen,  these  two  values  of  k  are  not  in  good  agreement, 
although  the  first  value  of  k  agrees  closely  with  the  mean  value 
of  k,  given  in  the  fourth  column  of  the  table. 
In  order  to  apply  the  equation, 

dx      ,  , 

5  =  k  (a  -  x)  , 

it  must  be  integrated. 

The  integration  of  this  equation  may.be  performed  as  follows:  — 

dx      ,  ,  N 

dt  =  k  (a  ~  x}> 

therefore, 

dx         ,   ,. 
-  =  k  dt. 
a  —  x 

Integrating,  we  have 

—  -  ---  I  kdt  =  constant  =  C, 
a-  x     J 

therefore, 

—  loge  (a  —  x)  —  kt  =  C. 

In  order  to  determine  C,  the  constant  of  integration,  we  make 
use  of  the  experimental  fact,  that  when  t  =  0,  x  =  0.  Substitut- 
ing these  values,  we  have 

—  loge  a  =  C. 
Consequently 

loge  a  -  loge  (a  -  x)  =  kt, 

or 


Passing  to  Briggsian  logarithms,  we  obtain 

i  log  -^~x=  0.4343  k.  (7) 


r  A,    *    3.'30?>  fA     f   eu-x- 

By  substituting  in  this  equation,  the  corresponding  values  of  a, 
a  —  x,  and  Z,  from  the  preceding  table,  the  values  of  k  given  in  the 
fourth  column  of  the  table  are  obtained. 


CHEMICAL  KINETICS  365 

The  equation, 

dx       ,  ,  N 

-=k(a-x), 

may  also  be  thrown  into  an  exponential  form,  as  follows: 
Since, 

1,          a 


we  may  write, 

7         ,       a  —  x 
-  kt  =  loge  —  — 

or 

a  —  x  =  ae~~tfj 
and 

x  =  a  (1  -  e-*).  (8) 

In  this  equation,  k  may  be  regarded  as  the  fraction  of  the  total 
amount  of  substance  decomposing  in  the  unit  of  time,  provided 
this  unit  is  so  small,  that  the  quantity  at  the  end  of  the  time 
is  only  slightly  different  from  that  at  the  beginning.  The  time 
required  for  one-half  of  the  substance  to  change,  is  known  as  the 
period  of  half-change,  T,  and  may  be  calculated  from  k  by  means 
of  the  equation, 

log  2  =  0.4343  k  T, 
therefore 

T  =  0.6943  jj;>  (9) 

or 

7  =  1.443  T.  (10) 

/c 

Reactions  in  which  only  one  mol  of  a  single  substance  undergoes 
change,  are  known  as  unimolecular  reactions,  or  reactions  of  the 
first  order.  In  a  unimolecular  reaction,  the  velocity  constant,  k, 
is  independent  of  the  units  in  which  concentration  is  expressed. 
If,  in  the  integrated  equation, 

kt  = 

/  becomes  infinite,  then  x  =  a.     In  other  words,  for  finite  values 
of  t,  x  must  always  remain  less  than  a  and  the  reaction  will  never 
proceed  to  completion. 
Another   unimolecular   reaction   which   has   been   thoroughly 


366  THEORETICAL  CHEMISTRY 

investigated,  is  the  hydrolysis  of  cane  sugar.  When  cane  sugar 
is  dissolved  in  water,  containing  a  small  amount  of  free  acid,  it  is 
slowly  transformed  into  d-glucose  and  d-fructose.  The  velocity 
of  the  reaction  is  very  small,  and  is  dependent  upon  the  strength 
of  the  acid  added.  The  progress  of  the  reaction  may  be  very 
easily  followed  by  means  of  the  polarimeter.  Cane  sugar  is 
dextro-rotatory,  while  d-fructose  rotates  the  plane  of  polariza- 
tion more  strongly  to  the  left  than  d-glucose  rotates  it  to. the  right. 
Therefore,  as  the  hydrolysis  proceeds,  the  angle  of  rotation  to  the 
right  steadily  diminishes  until,  when  the  reaction  is  complete,  the 
plane  of  polarization  will  be  found  to  be  rotated  to  the  left.  On 
this  account,  the  hydrolysis  of  cane  sugar  is  commonly  termed 
inversion,  and  the  molecular  mixture  of  d-fructose  and  d-glucose, 
constituting  the  product  of  the  reaction,  is  called  invert  sugar. 
Let  OQ  denote  the  initial  angle  of  rotation,  at  the  time  t  =  0, 
due  to  a  mols  of  cane  sugar,  let  <V  denote  the  angle  of  rotation, 
when  inversion  is  complete,  and  let  a  be  the  angle  of  rotation  at 
any  time  t',  then,  since  rotation  of  the  plane  of  polarization  is  pro- 
portional to  the  concentration  x,  the  amount  of  cane  sugar  in- 
verted, will  be 


Ci(\   —    Oi 

x  =  a  — 


In  the  equation, 

n  +  H20  <±  C6H1206 


representing  the  inversion  of  cane  sugar,  the  velocity  of  the 
reaction,  according  to  the  law  of  mass  action,  will  be  proportional 
to  the  molecular  concentrations  of  the  cane  sugar  and  the  water. 
Since  the  reaction  takes  place  in  the  presence  of  such  a  large  excess 
of  water,  its  effect  may  be  considered  to  be  constant.  The 
velocity  of  the  reaction,  is  then  proportional  to  the  active  mass 
of  the  sugar  alone,  or  in  other  words,  the  reaction  is  unimolecular. 
In  the  differential  equation,  expressing  the  velocity  of  a  unimolec- 
ular reaction, 


we  have 


dx      1  ,          N 
-  =  k(a-x), 


,       1,          a 
k  =7  log. 


t    &  'a-x' 


CHEMICAL  KINETICS 


367 


and,  since  a  and  x  are  measured  in  terms  of  angles  of  rotation  of 
the  plane  of  polarization,  we  have 


(11) 


The  following  table  gives  the  results  obtained  with  a  20  per  cent 
solution  of  cane  sugar  in  the  presence  of  0.5  molar  solution  of  lactic 
acid  at  25°  C. 


HYDROLYSIS  OF  CANE  SUGAR 


t  (minutes). 

a 

k 

0 

34°.  5 

1,435 

31°.  1 

0.2348 

4,315 

25°.  0 

0.2359 

7,070 

20°.  16 

0.2343 

11,360 

13°.  98 

0.2310 

14,170 

10°.  01 

0.2301 

16,935 

7°.  57 

0.2316 

19,815 

5°.  08 

0.2991 

29,925 

-  1°.65 

0.2330 

Inf. 

-10°.  77 

Bimolecular  Reactions.  When  two  substances  react,  and  the 
concentration  of  each  changes  at  the  same  rate,  the  reaction  is 
bimolecular,  or  of  the  second  order.  Let  a  and  b  represent  the  initial 
molar  concentrations  of  the  two  reacting  substances,  and  let  a* 
denote  the  amount  transformed  in  the  interval  of  time  £;  then 
the  velocity  of  the  reaction  will  be  expressed  by  the  equation, 


dx 

-     =  k  (a  -  x)  (b  -  x) 


(12) 


The  simplest  case  is  that  in  which  the  two  substances  are  present 
in  equivalent  amounts.  Under  these  conditions,  the  velocity 
equation  becomes 


=!(«-*)*. 


(13) 


Integrating  equation  (13),  we  have 


r  a  (a  -  x) 


(14) 


368  THEORETICAL  CHEMISTRY 

If  the  reacting  substances  are  not  present  in  equivalent  amounts, 
then  we  must  employ  the  equation 

-jt  =  k  (a  -  x)  (b  -  x). 
On  integration,  this  becomes 


Or,  passing  to  Briggsian  logarithms, 


The  value  of  k,  in  a  bimolecular  reaction,  is  not  independent  of 
the  units  in  which  the  concentration  is  expressed,  as  is  the  case 
with  a  unimolecular  reaction.  Suppose  that  a  unit  I/nth  of 
that  originally  selected  is  used  to  express  concentration,  then  the 
value  of  k  in  the  equation, 

1         x 

K    —  -    —,  -  r  > 

t  a  (a  —  x) 
becomes 

1  nx  1  x 


k'  = 


t    na  -  n(a  —  x)       t     na(a  —  x) 


Thus,  the  value  of  k  varies  inversely  as  the  numbers  expressing 
the  concentrations. 

As  an  illustration  of  a  bimolecular  reaction,  we  may  take  the 
hydrolysis  of  an  ester  by  an  alkali.     The  reaction, 

CH3COOC2H5  +  NaOH  <=»  CH3COONa  +  C2H5OH, 

has  been  studied  by  Warder,*  Reicher,  f  Arrhenius,  t  Ostwald  § 
and  others.  Arrhenius  employed,  in  his  experiments,  0.02  molar 
solutions  of  ester  and  alkali.  These  solutions  were  placed  in 
separate  flasks,  and  warmed  to  25°  C.  in  a  thermostat  maintained 
at  that  temperature;  equal  volumes  were  then  mixed  and,  at 

*  Berichte,  14,  1361  (1881). 

t  Lieb.  Ann.,  228,  257  (1885). 

J  Zeit.  phys.  Chem.,  i,  110  (1887). 

§  Jour,  prakt.  Chem.,  35,  112  (1887). 


CHEMICAL  KINETICS 


369 


frequent  intervals,  a  portion  of  the  reaction  mixture  was  removed 
and  titrated  rapidly  with  standard  acid.  The  accompanying 
table  contains  some  of  the  results  obtained : 

HYDROLYSIS  OF  ETHYL  ACETATE 


/  (minutes). 

a  —  x 

k 

0 

8.04 

4 

5.30 

0.0160 

6 

4.58 

0.0156 

8 

3.91 

0.0164 

10 

3.51 

0.0160 

12 

3.12 

0.0162 

Mean  0.0160 

The  numbers  in  the  second  column  of  the  table  represent  the 
concentrations  of  sodium  hydroxide  and  of  ethyl  acetate, 
expressed  in  terms  of  the  number  of  cubic  centimeters  of  standard 
acid  required  to  neutralize  10  cc.  of  the  reaction  mixture.  Owing 
to  the  high  velocity  of  the  reaction,  it  is  difficult  to  avoid  large 
experimental  errors,  nevertheless,  the  values  of  k,  given  in  the  third 
column  of  the  table,  will  be  observed  to  differ  very  slightly  from 
the  mean  value. 

Reicher  investigated  the  same  reaction,  when  the  reacting  sub- 
stances were  not  present  in  equivalent  proportions.  In  this  case, 
the  progress  of  the  reaction  was  followed  by  titrating  definite 
portions  of  the  reaction  mixture,  from  time  to  time,  the  excess  of 
sodium  hydroxide  being  determined  by  titrating  a  portion  of 
the  mixture  at  the  expiration  of  twenty-four  hours,  when  the 
ester  was  completely  hydrolyzed.  His  results  are  given  in  the 
following  table: 

HYDROLYSIS  OF  ETHYL  ACETATE 


t  (minutes). 

(alkali). 

b-x 

(ester). 

» 

47  03 

4.89 
11.36 
29.18 

T_f 

50.59 
42.40 
29.35 

14.  09 

35.67 
27.48 
14.43 

0.00093 
0.00094 
0.00092 

370  THEORETICAL  CHEMISTRY 

Reicher  also  studied  the  effect  of  different  bases  upon  the  veloc- 
ity of  the  reaction.  He  found,  for  strong  bases,  approximately 
equal  values  of  k,  but  for  weak  bases,  the  values  were  irregular 
and  smaller  than  those  obtained  with  the  more  completely  ionized 
bases.  Arrhenius  pointed  out,  that  the  hydrolyzing  power  of  a 
base  is  proportional  to  the  number  of  hydroxyl  ions  which  it 
yields.  Writing  the  equation  for  the  above  hydrolysis  in  terms 
of  ions,  we  have 

CH3COOC2H6  +  Na'  +  OH'  <=>  CH3COO'  +  Na'  +  C2H5OH. 

It  is  evident  from  this  equation,  that  all  bases  furnishing  the  same 
number  of  hydroxyl  ions  should  give  identical  values  of  k. 
Therefore,  we  may  modify  the  fundamental  differential  equation 
as  follows: 

^  =  k'a  (a  -  x)  (b-  x),  (17) 

where  a  is  the  degree  of  ionization  of  the  base. 

Trimolecular  Reactions.  Any  reaction  in  which  three  reacting 
molecules  are  directly  involved  is  trimolecular,  or  of  the  third  order. 
If  the  initial  molar  concentrations  of  the  reacting  substances  are 
denoted  by  a,  6,  and  c,  and  if  x  denotes  the  proportion  ^of  each 
which  is  transformed  in  the  interval  of  time  t,  the  velocity'of  the 
reaction  will  be  represented  by  the  differential  equation, 

-IT  =  k  (a  —  x)  (b  —  x)  (c  —  x).  (18) 

If  the  substances  are  present  in  equivalent  amounts,  the  equation 
becomes  ^ 


an  expression  which  is  much  less  difficult  to  integrate. 
On  integration,  equation  (19)  becomes 

1    x(2a-x) 
*-r2a'(a-s)V  (20) 

When  the  reacting  substances  are  not  taken  in  equivalent  amounts, 
the  integration  of  the  velocity  equation  gives, 


(0-6)  (b-c)  (c-o) 


CHEMICAL  KINETICS 


371 


In  a  trimolecular  reaction,  k  is  inversely  proportional  to  the  square 
of  the  original  concentration. 

A  typical  trimolecular  reaction  is  that  between  ferric  and 
stannous  chlorides.  This  reaction,  represented  by  the  following 
equation 

2  FeCl3  +  SnCl2  <=±  2  FeCl2  +  SnCl4, 

has  been  investigated  by  A.  A.  Noyes.*  Dilute  solutions  of  the 
reacting  substances  were  mixed  at  constant  temperature,  and 
definite  portions  of  the  reaction  mixture  were  removed,  at  meas- 
ured intervals  of  time,  and  titrated  for  ferrous  iron.  Before 
titrating  with  a  standard  solution  of  potassium  permanganate,  it 
was  necessary  to  decompose  the  stannous  chloride  present  with 
mercuric  chloride.  The  following  table  gives  the  results  obtained 
with  0.025  molar  solutions  of  ferric  chloride  and  stannous  chloride. 

REDUCTION  OF  FeCl3  BY  SnCU 


t  (minutes). 

a  —  x 

X 

k 

2.5 

0.02149 

0.00351 

113 

3 

0.02112 

0.00388 

107 

6 

0.01837 

0.00663 

114 

11 

0.01554 

0.00946 

116 

15 

0.01394 

0.01106 

118 

18 

0.01313 

0.01187 

117 

30 

0.01060 

0.01440 

122 

60 

0.00784 

0.01716 

122 

Mean  116 

Noyes  also  found,  that  the  velocity  of  the  reaction  is  accelerated 
more  by  an  excess  of  ferric  chloride  than  by  an  equal  excess  of 
stannous  chloride. 

Reactions  of  Higher  Orders.  Reactions  of  the  fourth,  fifth 
and  eighth  orders,  have  recently  been  investigated,  but  examples 
of  reactions  of  orders  higher  than  the  third  are  extremely  rare. 
This  fact  is  at  first  sight  surprising,  since  the  equations  of  many 
chemical  reactions  involve  a  large  number  of  molecules,  and  we 
should  naturally  expect  a  corresponding  increase  in  the. order  of 
such  reactions.  For  example,  the  reaction  represented  by  the 
equation, 

2  PH3  +  4  O2  =  P205  +  3  H2O, 

*  Zeit.  phys.  Chem.,  16,  546  (1895). 


372  THEORETICAL  CHEMISTRY 

involves  six  molecules  of  the  substances  initially  present,  and 
therefore,  we  should  infer  it  to  be  a  reaction  of  the  sixth  order. 
Kinetic  experiments  by  van  der  Stadt,  have  shown  it  to  be  a 
bimolecular  reaction,  the  velocity  of  reaction  being  proportional 
to  the  concentrations  of  the  phosphine  and  the  oxygen.  On 
allowing  the  gases  to  mix  slowly  by  diffusion,  it  was  discovered, 
that  the  reaction  actually  takes  place  in  several  successive  stages, 
the  first  stage  being  represented  by  the  equation  of  the  bimolec- 
ular reaction, 

PH3  +  O2  =  HPO2  +  H2. 

The  subsequent  changes,  involving  the  oxidation  of  the  products 
of  this  reaction,  take  place  with  great  rapidity. 

It  is  highly  probable,  that  the  equations  which  are  ordinarily  em- 
ployed to  represent  chemical  reactions,  really  represent  only  the 
initial  and  final  stages  of  a  series  of  relatively  simple  reactions. 
Larmor  *  has  shown,  that  when  chemical  reactions  are  considered 
from  the  molecular  standpoint,  the  bimolecular  reaction  is  the 
most  probable.  He  says,  "  Imagine  a  substance,  say  gaseous  for 
simplicity,  formed  by  the  immediate  spontaneous  combination  of 
three  gaseous  components  A,  B,  and  C.  When  these  gases  are 
mixed,  the  chances  are  very  remote  of  the  occurrence  of  the  sim- 
ultaneous triple  encounter  of  an  A,  a  B,  and  a  C,  which  would  be 
necessary  to  the  immediate  formation  of  an  ABC]  whereas,  if  ever 
formed,  it  would  be  liable  to  the  normal  chance  of  dissociating  by 
collisions;  it  would  thus  be  practically  non-existent  in  the  statistical 
sense.  But  if  an  intermediate  combination,  AB,  could  exist,  very 
transiently,  though  long  enough  to  cover  a  considerable  fraction 
of  the  mean  free  path  of  the  molecules,  this  will  readily  be  formed 
by  ordinary  binary  encounters  of  A  and  B,  and  another  binary 
encounter  of  A  B  with  C  will  now  form  the  triple  compound  ABC 
in  quantity." 

Determination  of  the  Order  of  a  Reaction.  It  has  been 
shown  in  the  foregoing  pages,  that  the  time  required  to  complete 
a  certain  fraction  of  a  reaction  is  dependent  upon  the  order  of 
the  reaction  in  the  following  manner: 

(1)  In  a  unimolecular  reaction,  the  value  of  k  is  independent  of 
the  initial  concentration; 

(2)  In  a  bimolecular  reaction,  the  value  of  k  is  inversely  pro- 
portional to  the  initial  concentration; 

*  Proc.  Manchester  Phil.  Soc.,  1908. 


CHEMICAL  KINETICS  373 

(3)  In  a  trimolecular  reaction,  the  value  of  k  is  inversely  pro- 
portional to  the  square  of  the  initial  concentration. 

Hence,  in  general,  in  a  reaction  of  the  nth  order,  the  value  of 
k  is  inversely  proportional  to  the  (n  —  1)  power  of  the  initial  con- 
centration. If  the  value  of  k  is  determined  with  definite  concen- 
trations of  the  reacting  substances,  and  then,  with  multiples  of 
those  concentrations,  the  order  of  the  reaction  can  be  determined, 
according  to  the  above  rules,  by  observing  the  manner  in  which  k 
varies  with  the  concentration. 

The  order  of  a  reaction  may  also  be  readily  determined  by 
means  of  a  graphic  method.  Thus,  to  determine  the  order  of  a 
reaction,  we  ascertain  by  actual  trial  which  one  of  the  following 
expressions,  in  which  C  denotes  concentration,  will  give  a  straight 
line  when  plotted  against  times  as  abscissas: 

(1)  log  C  —  reaction  unimolecular; 

(2)  1/C  —  reaction  bimolecular; 

(3)  1/C2  —  reaction  trimolecular; 

(4)  l/Cn  —  reaction  n  +  1  molecular. 

Complex  Reaction  Velocities.  Thus  far  we  have  considered 
the  velocity  of  reactions  which  are  practically  complete.  There 
are  numerous  cases,  however,  in  which  the  course  of  the  reaction 
is  complicated  by  such  disturbing  factors,  as  (1)  counter  reactions, 
(2)  side  reactions,  and  (3)  consecutive  reactions.  These  disturb- 
ing causes  will  now  be  considered. 

(1)  Counter  Reactions.  In  the  chemical  change  represented  by 
the  equation, 

CHsCOOH  +  C2H5OH  <=±  CH3COOC2H5  +  H20, 

the  speed  of  the  direct  reaction  steadily  diminishes,  owing  to  the 
ever-increasing  effect  of  the  reverse,  or  counter  reaction.  Ulti- 
mately, whe'n  two-thirds  of  the  acid  and  alcohol  are  decomposed, 
the  velocities  of  the  two  reactions  become  equal,  and  a  condition 
of  equilibrium  results.  Starting  with  1  mol  of  acid  and  1  mol 
of  alcohol,  and  letting  x  represent  the  amount  of  ester  formed, 
we  have 

^  =  k  (1  -  x?  -  k'x\  (22) 

When  equilibrium  is  attained, 

V    —  --   A 

Ke-p-1. 


374  THEORETICAL  CHEMISTRY 

By  observing  the  change  for  any  time  t,  we  have 

-  (23) 


Having  the  values  of  k/kf,  and  k  —  k',  the  velocity  constant,  fc, 
of  the  direct  reaction  can  be  determined.  The  value  of  k,  so 
obtained,  has  been  shown  by  Knoblauch  *  to  vary  in  those  reac- 
tions where  the  concentration  of  the  hydrogen  ion  changes. 

(2)  Side  Reactions.  When  the  same  substances  are  capable  of 
reacting  in  more  than  one  way,  with  the  formation  of  different 
products,  the  several  reactions  proceed  side  by  side.  Thus, 
benzene  and  chlorine  may  react  in  two  ways,  as  shown  by  the 
equations, 

(1)  C6HC  +  C12  =  C6H5C1  +  HC1, 
and 

(2) 


It  is  generally  possible  to  regulate  the  conditions  under  which 
the  substances  react,  so  as  to  promote  one  reaction  and  retard  the 
other. 

(3)  Consecutive  Reactions.  By  consecutive  reactions,  we  under- 
stand those  reactions,  in  which  the  products  of  a  certain  initial 
chemical  change,  react,  either  with  each  other,  or  with  the  original 
substances,  to  form  new  substances.  Attention  has  already  been 
called  to  the  fact,  that  many  of  our  common  chemical  equations 
really  represent  the  summation  of  a  number  of  consecutive  reac- 
tions. If  the  system  A,  is  transformed  into  the  system  C,  through 
an  intermediate  system  B,  then  we  shall  have  the  two  reactions, 

(1)  A-+B, 

and 

(2)  B  ->  C. 

If  reaction  (1)  should  have  a  very  much  greater  velocity  than 
reaction  (2),  then,  the  measured  velocity  of  the  change  from  A  to 
C,  will  be  practically  the  same  as  that  of  the  slower  reac- 
tion. This  fact  has  been  illustrated  by  means  of  the  following 
analogy,  due  to  James  Walker:  —  f  "  The  time  occupied  by  the 
transmission  of  a  telegraphic  message  depends  both  on  the  rate 

*  Zeit.  phys.  Chem.,  22,  268  (1897). 

f  Proc,  Roy.  Soc.,  Edinburgh,  22  (1898). 


CHEMICAL  KINETICS  375 

of  transmission  along  the  conducting  wire,  and  on  the  rate  of  prog- 
ress of  the  messenger  who  delivers  the  telegram;  but,  it  is  ob- 
viously this  last  slower  rate,  that  is  of  really  practical  importance 
in  determining  the  time  of  transmission."  The  saponification  of 
ethyl  succinate  may  be  taken  as  an  illustration  of  consecutive 
reactions.  This  reaction  proceeds  in  two  stages  as  follows: 

xCOQC2H5  /COOC2H5 


(1)     C2H4<Q  +  NaOH<=»C2H4<  +C2H^  IT 

XCOOC2H5 


<OOC2H5  xCOONa 

+  NaOH  <±  C2H4  <  +  C2H5Ot. 

OONa  XCOONa 

In  this  case,  the  product  of  the  first  reaction  reacts  with  one  of 
the  original  substances. 

Velocity  of  Heterogeneous  Reactions.  It  has  been  shown,  that 
when  a  solid,  such  as  calcium  carbonate,  is  dissolved  in  an  acid, 
the  rate  of  solution  is  dependent  upon  the  surface  of  contact 
between  the  solid  and  liquid  phases,  and  also  upon  the  strength 
of  the  acid.  If  the  surface  is  large  so  that  it  undergoes  relatively 
little  change  during  the  reaction,  it  may  be  considered  as  con- 
stant. If  S  represents  the  area  of  the  surface  exposed,  and  x 
denotes  the  amount  of  solid  dissolved  in  the  time  t,  the  velocity  of 
the  reaction  will  be  represented  by  the  differential  equation, 

§  =  kS  (a  -  x).  (24) 

Integrating  this  equation,  we  have 

^-.  (25) 


This  formula  has  been  tested  by  Boguski  *  for  the  reaction, 
CaC03  +  2  HC1  =  CaCl2  +  CO2  +  H2O, 

and  is  found  to  give  constant  values  of  k.  Furthermore,  Noyes 
and  Whitney  f  have  shown,  that  the  rate  of  solution  of  a  solid  in  a 
liquid  at  any  instant,  is  proportional  to  the  difference  between  the 
concentration  of  the  saturated  solution  and  the  concentration  of 
the  solution  at  the  time  of  the  experiment. 

*  Berichte,  9.  1646  (1876). 

t  Zeit.  phys.  Chem.,  23,  689  (1897). 


376 


THEORETICAL  CHEMISTRY 


Velocity  of  Reaction  and  Temperature.  It  is  a  well-estab- 
lished fact,  that  the  velocity  of  a  chemical  reaction  is  accelerated 
by  rise  of  temperature.  Thus,  the  rate  of  inversion  of  cane  sugar 
is  increased  about  five  times,  for  a  rise  in  temperature  of  30°.  It 
has  been  shown,  as  the  result  of  a  large  number  of  observations  on 
a  variety  of  chemical  reactions,  that  in  general  the  velocity  of  a 
reaction  is  doubled,  or  trebled,  for  an  increase  in  temperature  of 
10°  It  is  of  interest  to  note,  that  the  rate  of  development  of 
various  organisms,  such  as  yeast  cells,  the  rate  of  growth  of  the 
eggs  of  certain  fishes,  and  the  rate  of  germination  of  certain 
varieties  of  seeds,  is  either  doubled,  or  trebled,  for  a  rise  in  temper- 
ature of  10°.  Up  to  the  present  time,  no  wholly  satisfactory  for- 
mula, connecting  the  rate  of  reaction  with  the  temperature  has  been 
derived,  although  several  purely-empirical  expressions  have  been 
suggested.  Of  these  formulas,  the  most  widely  applicable  is  that 
proposed  by  van't  Hoff,  and  verified  by  Arrhenius.  If  k0  and  ki 
represent  the  velocity  constants  at  the  respective  temperatures, 
To  and  Ti,  then 

ki  =  ktf^ T*T*  (26) 

where  e  is  the  base  of  the  Naperian  system  of  logarithms,  and  A  is 
a  constant.  The  following  table  gives  the  calculated  and  observed 
values  of  k,  at  various  temperatures,  for  the  reaction, 

/NH2 
NH4CNO  <=>  OC<^ 


when  T  =  273  +  25°,  k  =  0.000227  and  A  =  11,700. 
VARIATION  OF   VELOCITY   CONSTANT   WITH   TEMPERATURE 


T, 
Degrees 

ib  (observed) 

k  (calculated) 

273  +  39 

0.00141 

0.00133 

273  +  50.1 

0.00520 

0.00480 

273  +  64.5 

0.0228 

0.0227 

273  +  74.7 

0.062 

0.0623 

273  +  80 

0.100 

0.105 

In  this  case,  the  agreement  between  the  observed  and  calculated 
values  is  all  that  could  be  desired. 


CHEMICAL  KINETICS 


377 


Influence  of  the  Solvent  on  the  Velocity  of  Reaction.    The 

velocity  of  a  chemical  reaction  varies  greatly  with  the  nature  of 
the  medium  in  which  it  takes  place.  This  subject  has  been 
studied  by  Menschutkin  *  who  has  collected  much  valuable  data, 
as  the  result  of  a  large  number  of  experiments,  on  the  velocity  of 
the  reaction  between  ethyl  iodide  and  triethylamine,  as  repre- 
sented by  the  equation, 

C2H5I  +  (C2H5)3N  =  (C2H5)4NL 

This  reaction  was  allowed  to  take  place  in  a  large  number  of 
different  solvents,  and  the  velocity  at  100°  was  measured.  A  few 
of  Menschutkin's  results  are  given  in  the  accompanying  table,  in 
which  A;  denotes  the  velocity  constant. 


VARIATION  OF  VELOCITY  CONSTANT  WITH 
REACTION  MEDIUM 


Medium. 

k 

Medium  . 

k 

Hexane  

0.00018 

Ethyl  alcohol  

0.0366 

Ethyl  ether 

0  000757 

Methyl  alcohol 

0  0516 

Benzene 

0  00584 

Acetone 

0  0608 

These  figures  show  that  the  velocity  of  the  reaction  is  greatly 
modified  by  the  nature  of  the  medium  in  which  it  takes  place,  the 
velocity  in  hexane  being  less  than  one  three-hundredth  of  that  in 
acetone.  It  is  of  interest  to  note,  that  there  is  an  approximate 
parallelism  between  the  values  of  k,  and  the  values  of  the  dielec- 
tric constant  of  the  different  media. 

Catalysis.  It  is  a  familiar  fact,  that  the  velocity  of  a  chemical 
reaction  is  frequently  greatly  accelerated  by  the  presence  of  a 
foreign  substance  which  apparently  does  not  participate  in  the 
reaction,  and  which  remains  unchanged  when  the  reaction  is 
complete.  For  example,  cane  sugar  is  inverted  very  slowly  by 
pure  water  alone,  but  when  a  trace  of  acid  is  added,  the  reaction 
is  greatly  accelerated.  A  substance  which  is  capable  of  exerting 
such  an  accelerating  action  is  termed  a  catalyst,  and  the  process  is 
known  as  catalysis.  Ostwald  likens  the  action  of  a  catalyst  to  that 
of  a  lubricant  on  a  machine,  —  it  helps  to  overcome  the  resistance 

*  Zeit.  phys.  Chem.,  6,  41  (1890). 


378  THEORETICAL  CHEMISTRY 

of  the  reaction.  If  the  velocity  of  a  reaction  is  represented  by 
an  equation  similar  to  that  expressing  Ohm's  law,  we  have 

,     .,       .         , .  driving  force 

velocity  of  reaction  = ^ 

resistance 

The  driving  force  is  the  same  thing  as  the  free  energy,  or  chemical 
affinity,  of  the  reacting  substances;  of  the  resistance  we  know 
practically  nothing.  The  velocity,  according  to  the  above  expres- 
sion, can  be  increased  in  either  of  two  ways,  viz.,  (1)  by  increas- 
ing the  driving  force,  or  (2)  by  diminishing  the  resistance.  It  is 
inconceivable  that  a  catalyst  can  exert  any  effect  upon  the  chem- 
ical affinity  of  the  reacting  substances,  hence  we  are  forced  to 
conclude,  that  its  action  must  be  confined. to  lessening  the  resist- 
ance. 

A  catalyst  has  been  defined,  by  Ostwald,  as  "  an  agent  which 
affects  the  velocity  of  a  chemical  reaction  without  appearing  in  the 
final  products  of  the  reaction."  While  this  definition  is  perhaps 
incomplete,  nevertheless,  it  embodies  the  outstanding  character- 
istics of  catalytic  action. 

The  Criteria  of  Catalysis.  The  following  criteria  characterize 
chemical  reactions  in  which  catalysis  plays  an  important  part. 

(1)  Positive  and   Negative  Catalysis.     According  to  Ostwald's 
definition,  a  catalytic  agent  may  exert  a  retarding,  as  well  as  an 
accelerating,  influence  on  the  velocity  of  a   chemical  reaction. 
A  catalyst  which  exerts  an  accelerating  influence,  is  known  as  a 
positive   catalyst,    whereas  a   catalyst   which    exerts   a   retarding 
influence,    is    called    a   negative   catalyst.     While    practically    all 
catalysts  are  positive  in  their  action,  there  are  a  few  which  are 
known  to  exert  an  opposite  influence.     Thus,  Bigelow  *  has  found 
that  the  rate  of  oxidation  of  sodium  sulphite  is  retarded  by  the 
presence  in  the  solution  of  as  little  as  one  one-hundred-and-sixty- 
thousandth  of  a  mol  of  mannite  per  liter. 

(2)  Unalterability  of  the  Catalyst.     While  a  catalyst   may  ac- 
tually participate  in  a  chemical  reaction,  it  will  reappear  at  the 
end  of  the  reaction,  unaltered  both  in  chemical  composition  and 
in  amount.     This  is  undoubtedly  the  most  striking  characteristic 
of  all  catalytic  agents.     In  this  connection,  it  should  be  stated,  that 
notwithstanding  the  fact  that  the  chemical  composition  of  a  cat- 
alyst remains  unchanged,  it  quite  often  happens  that  its  physical 

*  Zeit.  phys.  Chem.,  26,  493  (1898). 


CHEMICAL  KINETICS  379 

condition  undergoes  marked  alteration.  In  fact,  many  catalysts 
do  not  attain  to  their  maximum  efficiency  until  such  alteration, 
or  "  ageing,"  has  occurred. 

(3)  Amount   of  Catalyst   Required.     The   amount   of   catalyst 
required  to  promote  the  transformation  of  a  large  mass  of  reacting 
material  is  generally  exceedingly  small.     The  amount  of  catalyst 
also  remains  constant,  unless  interfering  side-reactions  take  place. 
In  certain  cases,  the  amount  of  catalyst  is  found  to  increase  as  the 
reaction  proceeds,  while  in  other  cases,  the  amount  of  catalyst  is 
found  to  decrease.     This  has  been  shown  to  be  due  to  the  fact, 
that  the  catalyst  is  actually  being  formed,  or  absorbed,  as  the  reac- 
tion proceeds.     Thus,  when  metallic  copper  is  dissolved  in  nitric 
acid,  the  reaction  proceeds,  slowly  at  first,  and  then  after  a  short 
interval,  the  speed  of  the  reaction  is  greatly  augmented.  .The 
acceleration  is  due  to  the  catalytic  action  of  the  nitric  oxide 
evolved.     This  phenomenon  is  known  as  autocatalysis.     In  reac- 
tions where  autocatalysis  occurs,  the  velocity  increases  with  the 
time,  until  a  certain  maximum  value  is  reached,  after  which,  the 
velocity  steadily  diminishes.     In  ordinary  reactions,  the  initial 
velocity  is  the  greatest. 

As  a  generalization,  to  which  there  are  numerous  exceptions,  it 
may  be  stated  that  reaction  velocity  is  approximately  propor- 
tional to  the  amount  of  catalyst  present. 

(4)  Role  of  Catalyst  in  Initiating  Reaction.     As  to  whether,  or 
not,  a  catalyst  is  capable  of  initiating  a  chemical  reaction,  it  is 
difficult   to   decide.     Take,   for  example,   the  reaction   between 
hydrogen  and  oxygen,  which  does  not  appear  to  take  place  at  all 
under  ordinary  conditions  of  temperature,  unless  some  catalyst, 
such  as  finely  divided  platinum,  is  present.     Ostwald  contends, 
that  the  reaction  actually  occurs  at  ordinary  temperatures,  but 
that  the  velocity  is  too  small  to  admit  of  measurement.     Others 
maintain,  that  no  reaction  occurs  under  these  conditions,  and  that 
the  introduction  of  a  catalyst  is  necessary  in  order  to  initiate  the 
combination  of  the  two  gases. 

(5)  Influence  of  Catalyst  upon  Equilibrium.     As  we  have  seen, 
the  final  state  of  equilibrium  in  a  chemical  reaction,  depends  solely 
on  the  ratio  of  the  velocities  of  the  direct  and  inverse  reactions. 
Since  this  equilibrium  is  not  disturbed  by  the  introduction  of  a 
catalyst,  it  follows,  that  the  velocities  of  the  two  reactions  must  be 
altered  to  the  same  extent.     Furthermore,  the  amount  of  energy 


380  THEORETICAL  CHEMISTRY 

transformed  in  a  chemical  reaction,  represents  the  difference  be- 
tween the  amounts  of  energy  stored  in  the  system  in  its  initial 
and  final  states.  Since  the  catalyst  introduces  no  energy,  it  is 
evident,  that  it  cannot  exert  any  influence  on  the  final  state  of 
equilibrium. 

(6)  Influence  of  Foreign  Substances  on  Catalyst.     The  activity 
of  a  catalyst  is  appreciably  altered  by  the  presence  of  extremely 
minute  amounts  of  foreign  substances.     In  some  instances,  the 
activity  is  diminished,  and  in  others,  it  is  increased.     Foreign  sub- 
stances which  tend  to  inhibit  catalytic  activity  are  known  as 
poisons,  while  substances  which  tend  to  enhance  the  activity  are 
called  promoters.     The  presence  of  the  merest  trace  of  a  poison 
is  often  sufficient  to  cause  the  complete  paralysis  of  a  catalyst. 
For  example,  in  the  manufacture  of  sulphuric  acid  by  the  familiar 
contact  process,  the  presence  of  a  very  minute  amount  of  arsenic 
in  the  sulphur  dioxide  has  been  found  sufficient  to  destroy  com- 
pletely the  activity  of  the  platinum-asbestos  catalyst.     On  the 
other  hand,  the  mixture  of  minute  amounts  of  certain  promoters, 
such  as  finely  divided  oxides,  with  practically  any  of  the  ordinary 
metallic   catalysts,  is  found   greatly  to   enhance  their  activity. 
Thus,  in  the  Haber  process  for  the  synthesis  of  ammonia  from  its 
elements,  the  addition  of  a  trace  of  certain  metallic  salts  to  the 
finely  divided  iron  used  as  a  catalyst,  produces  an  enormous  in- 
crease in  its  activity. 

(7)  Specificity  of  Catalytic  Action.     While  it  is  probable  that 
every  chemical  reaction  is  more  or  less  susceptible  to  catalytic 
influence,  and  also  that  practically  every  known  substance  is 
capable  of  functioning  as  a  catalyst,  nevertheless,  it  is  certain  that 
each  reaction  requires  a  particular  catalyst,  or  group  of  catalysts, 
%o  modify  its  velocity.     In  other  words,   catalytic   activity  is 
specific. 

Mechanism  of  Catalysis.  As  to  the  cause  of  catalytic  action 
very  little  is  known.  In  fact,  it  is  more  reasonable  to  suppose  that 
the  mechanism  of  catalysis  varies  with  the  nature  of  the  reaction 
and  the  nature  of  the  catalyst,  than  to  conceive  all  catalytic  effects 
to  be  traceable  to  a  common  origin.  One  of  the  earliest  hypothe- 
ses as  to  the  mechanism  of  catalysis  was  put  forward  by  Liebig. 
He  suggested  that  the  catalyst  sets  up  intramolecular  vibrations 
which  assist  chemical  reaction.  The  vibration  theory  was  grad- 
ually abandoned  as  its  inadequacy  came  to  be  recognized.  Of  the 


CHEMICAL  KINETICS  381 

many  explanations  which  have  been  offered  to  account  for  cataly- 
tic acceleration,  that  involving  the  formation  of  hypothetical 
intermediate  compounds  with  the  catalyst  has  been  accepted  with 
the  greatest  favor.  Thus,  if  a  reaction  represented  by  the  equa- 
tion, 

A  +  B  =  AB, 

takes  place  very  slowly  under  ordinary  conditions,  it  is  possible 
to  accelerate  its  velocity  by  the  addition  of  an  appropriate  cat- 
alyst, C.  According  to  the  theory  of  intermediate  compounds, 
the  catalyst  is  supposed  to  act  in  the  following  manner: 

(1)  A  +  C  =  AC, 

(2)  AC  +  B  =  AB  +  C. 

As  will  be  seen,  the  catalyst  is  regenerated  in  the  second  stage 
of  the  reaction.  A  catalyst  which  functions  in  this  manner  is 
known  as  a  Carrier.  In  1806,  Clement  and  Desormes  suggested 
that  the  action  of  nitric  oxide  in  promoting  the  oxidation  of  sul- 
phur dioxide,  in  the  manufacture  of  sulphuric  acid,  was  purely 
catalytic.  As  is  well  known,  the  rate  of  the  reaction  represented 
by  the  equation, 

2  SO2  +  O2  =  2  SO3, 

is  very  slow.  The  accelerating  action  of  nitric  oxide  on  the 
reaction  may  be  represented  in  the  following  manner: 

(1)  2  NO  +  02  =  2  NO2, 
and 

(2)  S02  +  N02  =  S03  +  NO. 

This  explanation,  first  offered  by  Clement  and  Desormes,  of  the 
part  played  by  the  oxides  of  nitrogen  in  the  synthesis  of  sulphuric 
acid,  is  still  regarded  as  the  most  plausible  although  it  is  apparent 
that  it  is  far  from  complete. 

Another  explanation  of  the  mechanism  of  catalysis  is  based 
upon  the  phenomenon  of  adsorption.  According  to  this  theory, 
the  increase  in  concentration  which  results  from  the  adsorption 
of  the  components  of  a  reaction  mixture,  by  a  finely  divided  cat- 
alyst, is  the  cause  of  the  acceleration  in  reaction  velocity.  In 
the  immediate  neighborhood  of  the  catalyst,  where  the  concen- 
tration of  the  reacting  substances  is  high,  the  rate  of  reaction  will 
undoubtedly  be  greater  than  throughout  the  mass  of  the  reaction 
mixture,  where  the  concentration  is  much  lower,  and  where  the 
distribution  of  the  molecules  is  far  less  favorable  to  reaction. 


382  THEORETICAL  CHEMISTRY 

For  further  information  on  the  mechanism  of  catalytic  action, 
the  student  is  referred  to  an  exhaustive  review,  by  Bancroft,* 
of  the  various  theories  which  have  been  advanced  in  explanation 
of  catalytic  phenomena. 

Catalytic  Action  of  Water.  Water,  seemingly,  functions  as  a 
catalyst  in  a  large  number  of  chemical  reactions.  Thus,  it  has 
been  shown  by  Baker,  f  that  no  reaction  occurs  when  hydrochloric 
acid  and  ammonia  are  brought  together  in  a  perfectly  dry  condi- 
tion, but  that  when  a  minute  trace  of  water  is  introduced,  the 
reaction  proceeds  rapidly  to  completion. 

Some  Applications  of  Catalysis.  A  few  of  the  many  applica- 
tions of  catalysis  in  the  realm  of  industrial  chemistry  should  be 
briefly  mentioned,  in  order  that  the  great  practical  importance  of 
this  interesting  subject  may  receive  sufficient  emphasis. 

(1)  The   Deacon   Process.     In    1888,  Deacon   and   Hurter,  by 
passing  a  mixture  of  hydrochloric  acid  gas  and  oxygen  over  pum- 
ice, impregnated  with  cupric  chloride  as  a  catalyst,  sliowed  that 
the  velocity  of  the  reaction, 

4  HC1  +  O2  =  2  H20  +  2  C12, 

can  be  sufficiently  accelerated  to  render  its  employment  entirely 
practicable  in  the  production  of  chlorine  on  the  commercial  scale. 
The  catalyst  is  very  sensitive  to  such  poisons  as  sulphur  dioxide, 
sulphur  trioxide  and  arsenic.  Water  also  has  been  found  to  inter- 
fere with  the  reaction,  so  that  it  is  necessary  that  the  two  gases 
shall  be  thoroughly  dried  before  being  allowed  to  enter  the  reac- 
tion chamber. 

(2)  The  Haber  Process.     Although  the  affinity  of  nitrogen  for 
hydrogen  is  known  to  be  so  feeble  that,  at  the  temperature  of  the 
electric   spark,   and  under  atmospheric   pressure,   practically  no 
ammonia  is  formed,  yet  by  a  careful  study  of  the  conditions  gov- 
erning the  equilibrium  represented  by  the  equation, 

N2  +  3  H2  <=»  2  NH3, 

and  by  the  employment  of  proper  catalytic  agents,  Haber  succeeded 
in  producing  ammonia  on  a  commercial  scale  by  means  of  this  re- 
action. At  1000°,  ammonia  may  be  regarded  as  completely  dis- 

*  Trans.  Am.  Electrochem.  Soc.,  37,  233  (1920). 
t  Ibid.  65,  611  (1894);  73,  422  (1898). 


CHEMICAL  KINETICS  383 

sociated,  while  at  550°,  and  under  atmospheric  pressure,  only 
about  0.08  per  cent  of  ammonia  is  formed.  Increase  of  pressure, 
however,  displaces  the  equilibrium,  in  accordance  with  the  prin- 
ciple of  Le  Chatelier,  so  that  more  of  the  system  occupying  the 
smaller  volume,  namely  ammonia,  is  formed.  Thus,  at  550°, 
and  under  a  pressure  of  200  atmospheres,  the  amount  of  ammonia 
formed  is  nearly  12  per  cent.  Despite  the  fact  that  this  percent- 
age is  so  small,  this  amount  of  ammonia  can  be  removed  as  am- 
monia of  crystallization  by  ammonium  nitrate,  at  a  low  tempera- 
ture, and  fresh  portions  of  hydrogen  and  nitrogen  introduced  into 
the  reaction  chamber,  thus  rendering  the  process  continuous. 
Various  catalysts  have  been  employed,  but  pure  iron  appears  to 
be  the  most  efficient.  With  this  catalyst  and  suitable  promoters, 
it  is  claimed  that  the  process  can  be  conducted  efficiently  at  pres- 
sures as  low  as  50  atmospheres.  Here  again,  the  catalyst  is  very 
susceptible  to  the  presence  of  poisons  such  as  sulphur,  selenium, 
tellurium,  arsenic  etc.  It  is  stated  that  0.01  per  cent  of  sulphur 
is  sufficient  to  destroy  completely  the  catalytic  activity  of  the 
iron  catalyst.  The  Haber  process  played  an  important  part  in 
providing  Germany  with  fixed  nitrogen  during  the  recent  war. 

(3)  Sabatier's  Hydrogenation  Process.  Toward  the  latter  part 
of  the  nineteenth  century,  Sabatier  and  Senderens*  discovered 
that  reduced  nickel  will  bring  about  the  combination  of  hydrogen 
with  ethylene,  or  acetylene,  with  the  formation  of  ethane,  as 
shown  by  the  equations, 

C2H4  +  H2    = 

-  2  H2  = 


In  this  method  of  hydrogenation,  the  vapor  of  the  substance  to 
be  reduced  is  mixed  with  hydrogen  and  passed  over  a  specific 
catalyst,  which  is  maintained  at  a  temperature  ranging  from  150° 
to  300°.  The  process  is  not  only  extremely  rapid,  but  also  requires 
a  minimum  amount  of  attention.  Furthermore,  the  yields  are 
relatively  high,  provided  proper  care  is  taken  to  maintain  the 
temperature  within  the  prescribed  limits  necessary  to  minimize 
the  formation  of  subsidiary  products.  Although  nickel  is  the 
most  efficient  catalyst,  other  metals,  such  as  cobalt,  platinum, 
iron  and  copper,  may  be  used.  Whatever  catalyst  is  employed, 

*  Compt.  rend.  124,  1358  (1897);   128,  1173  (1899). 


384  THEORETICAL  CHEMISTRY 

it  is  absolutely  essential  that  great  care  be  taken  in  its  prepara- 
tion, since  the  success  of  the  method  is  largely  dependent  upon 
the  condition  of  the  metal.  The  catalyst  is  very  susceptible  to 
various  poisons,  particularly  chlorine,  sulphur,  arsenic  and  phos- 
phorus and  their  compounds. 

The  process  has  found  numerous  applications,  notably  in  the 
preparation  of  artificial  perfumes.  It  also  has  proven  of  value  in 
the  synthesis  of  cyclohexanol  and  p-methylcyclohexanol,  from 
phenol  and  p-cresol,  respectively.  Both  of  these  substances  are 
used  in  the  preparation  of  isoprene  and  butadiene,  compounds 
of  importance  in  connection  with  the  synthesis  of  artificial  rub- 
ber. Undoubtedly  the  most  important  industrial  application  of 
the  hydrogenation  process  is  to  the  conversion  of  unsaturated 
aliphatic  acids,  such  as  oleic  acid,  into  saturated  acids,  such  as 
stearic  acid. 

(4)  Sabatier's  Dehydrogenation  Process.     Sabatier  and  his  asso- 
ciates also  showed,  that  at  a  given  temperature,  either  water,  or 
hydrogen,  may  be  split  off  from  alcohols,  depending  upon  the 
nature  of  the  catalyst  employed.     Such  finely  divided  metals 
as  copper,  cobalt,  nickel,  iron,  platinum  and  palladium,  together 
with  such  anhydrous  oxides  as  the  lower  oxides  of  manganese,  tin, 
uranium,  molybdenum  and  vanadium,  have  been  found  to  func- 
tion as  catalysts  in  the  dehydrogenation  process.     The  dehydro- 
genation  of  both  primary  and  secondary  alcohols  is  most  readily 
brought  about  by  the  use  of  reduced  copper  as  a  catalyst.     On 
passing  an  aliphatic  alcohol  over  the  latter,  at  a  temperature  of 
200°  to  300°,  a  yield  of  fully  50  per  cent  aldehyde,  or  ketone,  may 
reasonably  be  expected. 

The  chief  technical  application  of  this  process  is  to  the  produc- 
tion of  formaldehyde  from  methyl  alcohol.  In  this  process,  air 
is  drawn  through  methyl  alcohol,  maintained  at  a  temperature 
suitable  for  saturation,  and  the  mixture  is  then  passed  into  a  reac- 
tion chamber,  over  copper  gauze,  as  a  catalyst.  The  reaction 
which  occurs  may  be  represented  by  the  equations, 

2  CH3OH  +  O2  =  2  HCHO  +  4  H  +  O2  =  2  HCHO  +  2  H2O. 

(5)  Removal  of  Carbon  Monoxide  from  the  Air.     The  presence 
of  varying  amounts  of  carbon  monoxide  in  the  air,  due  to  the  in- 
complete combustion  of  solid  and  liquid  fuels,  together  with  its 
presence  in  the  products  of  combustion  of  smokeless  powder,  renders 


CHEMICAL  KINETICS  385 

the  development  of  adequate  means  for  its  elimination  of  the  utmost 
importance,  alike  in  times  of  peace  and  in  times  of  war.  A  method 
for  its  elimination  has  recently  been  devised  by  Lamb,  Bray  and 
Frazer.*  After  much  experimentation,  it  was  found  that  when 
a  mixture  of  air,  and  carbon  monoxide  is  passed,  at  ordinary  tem- 
peratures, over  a  catalyst  consisting  of  50  per  cent  MnO2,  30  per 
cent  CuO,  15  per  cent  Co2O3,  and  5  per  cent  Ag2O,  the  carbon 
monoxide  is  completely  oxidized  to  carbon  dioxide.  This  cataly- 
tic mixture,  called  "  hopcalite  "  by  its  discoverers,  has  been 
shown  to  be  capable  of  acting  indefinitely  against  any  concentra- 
tion of  carbon  monoxide,  provided  the  air  is  dry. 

(6)  The  Welsbach  Mantle.  The  ignited  Welsbach  incandescent 
gas  mantle  consists  of  a  fragile  web  of  the  oxides  of  thorium  and 
cerium.  Pure  thoria  gives  relatively  little  light,  but  if  progres- 
sive amounts  of  ceria  are  added,  the  luminosity  gradually  increases 
and  attains  maximum  brightness  when  1  per  cent  of  ceria  has  been 
added.  It  is  believed,  that  this  increase  in  luminosity  is  due  to  the 
alternate  oxidation  and  reduction  of  the  ceria,  as  indicated  by  the 
equation, 

02  +  2  Ce2O3  +±  4  Ce02. 

The  presence  of  1  per  cent  of  ceria  in  the  mixture  is  sufficient  to 
insure  maximum  velocity  of  surface  combustion. 

The  R61e  of  Ferments  and  Enzymes  in  Catalysis.  It  is  appar- 
ent, that  catalysis  is  of  great  importance  in  connection  with  many 
industrial  processes  as  well  as  in  the  field  of  pure  chemistry.  The 
majority  of  the  reactions  occurring  within  living  organisms  have 
likewise  been  found  to  be  accelerated  catalytically  by  unorganized 
ferments,  or  enzymes.  Thus,  before  the  process  of  digestion  can 
proceed,  starch  must  be  changed  into  sugar.  This  transforma- 
tion is  accelerated  by  an  enzyme  called  ptyalin  occurring  in  the 
saliva,  and  by  other  enzymes  found  in  the  pancreatic  juice.  The 
digestion  of  albumen  is  hastened  by  the  enzymes,  pepsin  and 
trypsin.  As  a  rule,  each  enzyme  acts  catalytically  on  just  one 
reaction,  or  in  other  words,  the  catalytic  action  of  enzymes  is  spe- 
cific. Enzymes  are  very  sensitive  to  traces  of  certain  toxic  sub- 
stances such  as  hydrocyanic  acid,  iodine,  and  mercuric  chloride. 

An  interesting  series  of  experiments  by  Bredig  f  on  the  cataly- 

*  Jour.  Ind.  Eng.  Chem.,  12,  213  (1920). 
f  Zeit.  phys.  Chem.,  31,  258  (1899). 


386  THEORETICAL  CHEMISTRY 

tic  action  of  colloidal  metals,  established  the  fact  that  these  sub- 
stances resemble  the  enzymes  very  closely  in  their  behavior. 
Thus,  they  are  "  poisoned  "  by  the  same  substances  which  inhibit 
the  activity  of  the  enzymes,  and  they  show  the  same  tendency  to 
recover  when  the  amount  of  the  poison  does  not  exceed  a  certain 
limiting  value.  Because  of  this  close  similarity,  Bredig  called  the 
colloidal  metals  inorganic  ferments. 

Among  the  numerous  applications  of  ferments  in  chemical  tech- 
nology, we  shall  limit  ourselves  to  the  mention  of  but  a  single 
example.  The  increased  demand  for  acetone  and  fusel  oil,  as 
solvents,  has  led  to  the  development,  by  Fernbach,  of  an  interesting 
fermentation  process  for  the  production  of  these  substances.  The 
Fernbach  process  may  be  outlined  as  follows  :  —  Corn,  or  any 
similar  material  rich  in  carbohydrates,  is  sterilized  at  130°,  made 
into  a  wort  with  water,  and,  after  adding  a  small  amount  of  de- 
graded yeast,  the  mixture  is  again  sterilized.  When  cool,  an 
anerobic  fermentation  process  is  conducted  by  means  of  an  appro- 
priate ferment,  at  a  temperature  of  about  30°.  The  mass  is  then 
distilled,  when  a  yield  of  from  33  to  50  per  cent  of  acetone  and 
fusel  oil,  calculated  on  the  basis  of  the  amount  of  carbohydrate 
used,  is  obtained. 

REFERENCES 

Chemical  Statics  and  Dynamics.     Mellor.     Chapters  II  to  VII  incl.,  also  X 

and  XL 

Catalysis  and  its  Industrial  Applications.     Jobling. 
Catalysis  in  Theory  and  Practice.     Rideal  and  Taylor. 

PROBLEMS 

1.  When  a  solution  of  dibromsuccinic  acid  is  heated,  the  acid  decom- 
poses into  brom-maleic  acid  and  l^ydrobromic  acid  according  to  the  equa- 

tion, \T 

CHBr-COOH     CH-COOH 


|  -fc  HBr. 

CHBr-COOH     C 


HBr-COOH     CBr-COOH 


At  50°,  the  initial  titre  of  a  definite  volume  of  the  solution  was 
To  =  10.095  cc.  of  standard  alkali.  After  t  minutes,  the  titre  of  the  same 
volume  of  solution  was  Tt  cc.  of  standard  alkali. 

t  0  214  380 

Tt  10.09  10.37  10.57 

(p.o<         -53 


CHEMICAL  KINETICS 


387 


(a)  Calculate  the  velocity-constant  of  the  reaction. 

(b)  After  what  time  is  one-third  of  the  dibromsuccinic  acid  decom- 
posed? Ans.  (a)  0.00026;  (b)  1559  min. 

2.  From  the  following  data  show  that  the  decomposition  of  H202  in 
aqueous  solution  is  a  unimolecular  reaction: 


Time  in  minutes 
n 


0 

22.8 


10 
13.8 


20 
8.25  cc. 


n  is  the  number  of  cubic  centimeters  of  potassium  permanganate  re- 
quired to  decompose  a  definite  volume  of  the  hydrogen  peroxide  solu- 
tion. 

3.  In  the  saponification  of  ethyl  acetate  by  sodium  hydroxide  at  10°, 
y  cc.  of  0.043  molar  hydrochloric  acid  were  required  to  neutralize  100  cc. 
of  the  reaction   mixture,  t  minutes   after  the   commencement  of  the 

reaction, 

&)          \e\  K}  (y 

t  0  4.89        10.37        28.18        infinity 

y  61.95          50.59        42.40        29.35        14.92 

/&iC~          /^  *?  /         VS 

Calculate  the  velocity-constant,  when  the  concentrations  are  expressed 
in  mols  per  liter.  Ans.  Mean  value  of  k  =  2.38. 

4.  The  velocity-constant  of  formation  of  hydriodic  acid  from  its  ele- 
ments is  0.00023;   the  equilibrium  constant  at  the  same  temperature  is 
0.0157.    What  is  the  velocity-constant  of  the  reverse  reaction? 

Ans.  0.0146. 

5.  Determine  the  order  of  the  following  reaction: 

6  FeCl2  +  KC103  +  6  HC1  =  6  FeCl3  +  KC1  +  3  H2O. 

When  the  initial  concentration  of  the  reacting  substances  is  0.1,  the 
changes  in  concentration  at  successive  times  are  as  follows: 


Time  (minutes). 

Change  in  Con 
centration. 

5 

0  0048 

15 

0.0122 

35 

0.0238 

60 

0  0329 

110 

0.0452 

170 

0.0525 

6.  At  395°  potassium  chlorate  is  supposed  to  decompose  according 
to  the  equation, 


4KC103  =KC1+3KC104. 


388  THEORETICAL  CHEMISTRY 

In  studying  this  reaction,  Scobai  obtained  the  following  data: 


Time 

Amount  of  KClOs  Trans- 
formed 

9 

0.0250 

24 

0.0373 

48 

0.0720 

72 

0.0954 

96 

0.1130 

120 

0.1211 

144 

0.1274 

Determine  the  order  of  the  reaction. 

7.  Calculate  the  relative  rates  at  which  a  crystalline  solid  will  dissolve 
in  its  own  solution,  (a)  when  the  solution  is  40  per  cent  saturated,  and 
(b)  when  it  is  90  per  cent  saturated. 

8.  The  velocity  constants  in  the  reaction  represented  by  the  equation 

2  AsH3  =  2  As  +  3  H2, 

are  k  =  0.00035  at  256°,  and  k  =  0.0034  at  367°.    Calculate  the  value 
of  the  velocity  constant  at  400°. 


CHAPTER  XV 
ELECTRICAL  CONDUCTANCE 

Historical  Introduction.  In  a  book  of  this  character  it  is 
impossible  to  give  anything  like  a  complete  historical  sketch  of 
electrochemistry.  Before  entering  upon  an  outline  of  this  inter- 
esting division  of  theoretical  chemistry,  however,  it  is  desirable 
to  consider  very  briefly  a  few  of  the  theories  which  have  played  a 
prominent  part  in  the  development  of  our  modern  views  concern- 
ing electrochemical  phenomena. 

While  the  early  observations  of  Beccaria  and  others  pointed 
to  the  probability  of  the  existence  of  some  relation  between 
chemical  and  electrical  phenomena,  it  was  not  until  the  begin- 
ning of  the  nineteenth  century  that  the  science  of  electrochem- 
istry had  its  birth.  The  epoch-making  discovery,  by  Volta,  of 
a  means  of  obtaining  electrical  energy  from  chemical  energy, 
gave  the  initial  impulse  to  all  the  brilliant  discoveries  and  in- 
vestigations upon  which  the  modern  science  of  electrochem- 
istry is  based.  The  apparatus  devised  by  Volta,  known  as 
the  voltaic  pile,  consisted  of  disks  of  zinc  and  silver  placed 
alternately  over  one  another,  the  silver  disk  of  one  pair  being 
separated  from  the  zinc  disk  of  the  next,  by  a  piece  of  blot- 
ting paper  moistened  with  brine.  Such  a  pile,  if  composed  of 
a  sufficient  number  of  pairs  of  disks,  will  produce  electricity 
enough  to  give  a  shock,  if  the  top  and  bottom  disks,  or  wires 
connected  with  them,  be  touched  with  the  moist  fingers.  This 
discovery  placed  in  the  hands  of  the  investigator  a  source  of  elec- 
tricity by  means  of  which  experiments  could  be  performed  which 
had  hitherto  been  impossible. 

Shortly  after  the  discovery  of  the  voltaic  pile,  Nicholson  and 
Carlisle*  effected  the  decomposition  of  water,  and  Davyf  isolated 
the  alkali  metals.  As  a  result  of  these  experiments,  Davy  was  led 
to  formulate  his  electrochemical  theory.  According  to  this  theory, 

*  Nich.  Jour.,  4,  179  (1800). 

t  Ibid.,  4,  275,  326  (1800);  Gilb.  Ann.,  7,  114  (1801).   . 
389 


390  THEORETICAL  CHEMISTRY 

the  atoms  of  different  substances  acquire  opposite  electrical  charges 
by  contact,  and  thus  mutually  attract  each  other.  If  the  differ- 
ences between  the  charges  are  small,  the  attraction  will  be  insuffi- 
cient to  cause  the  atoms  to  leave  their  former  positions;  if  it  is 
great,  a  rearrangement  of  the  atoms  will  occur,  and  a  chemical 
compound  will  be  formed.  In  terms  of  this  theory,  electrolysis 
consists  in  a  neutralization  of  the  charges  upon  the  atoms. 

The  theory  of  Davy  was  soon  superseded  by  that  of  Berzelius.* 
According  to  the  latter  theory,  every  atom  is  charged  with  both 
kinds  of  electricity,  which  exist  upon  the  atoms  in  a  polar  arrange- 
ment, the  electrical  behavior  of  the  atom  being  determined  by  the 
kind  of  electricity  which  is  in  excess.  Chemical  attraction  is  thus 
ascribed  to  the  electrical  attraction  of  oppositely-charged  atoms. 
Since  each  atom  is  endowed  with  both  positive  and  negative  elec- 
trification, one  charge  being  in  excess,  it  follows  that  the  com- 
pound formed  by  the  union  of  two  or  more  atoms  will  be  positively 
or  negatively  charged,  according  to  whichever  charge  remains 
unneutralized  after  the  atoms  have  combined.  Two  compounds, 
the  one  charged  positively  and  the  other  negatively,  may  thus 
in  turn  combine,  a  more  complex  compound  being  formed. 
Shortly  after  Berzelius  formulated  his  theory,  it  became  the  sub- 
ject of  much  discussion  and  was  severely  criticized.  Thus,  it  was 
pointed  out,  that  if  chemical  combination  results  from  the  neutral- 
ization of  oppositely-charged  atoms,  then  as  soon  as  the  charges 
have  become  equalized,  there  will  no  longer  exist  any  attractive 
force,  and  the  compound  will  again  decompose.  This  objection 
was  easily  overcome  by  assuming,  that  as  soon  as  the  union  be- 
tween the  atoms  is  broken,  they  will  again  acquire  their  original 
charges,  and  in  consequence,  will  recombine.  In  other  words,  a 
chemical  compound  is  to  be  regarded  as  existing  in  a  state  of 
unstable  equilibrium. 

Another,  and  apparently  insurmountable  objection  to  the  theory, 
resulted  from  the  exceptions  presented  by  acetic  acid  and  some  of 
its  substitution  products.  According  to  the  theory  of  Berzelius, 
chemical  combination  is  entirely  dependent  upon  the  nature  of 
the  electrical  charges  residing  on  the  atoms.  From  this  state- 
ment it  follows,  that  the  properties  of  a  chemical  compound  must 
be  a  function  of  the  electrical  charges  upon  the  atoms  of  its  con^ 
stituents.  It  was  shown,  that  when  the  three  hydrogen  atoms 
*  Gilb.  Ann..  27,  270  (1807). 


ELECTRICAL  CONDUCTANCE  391 

of  the  methyl  group  in  acetic  acid  are  successively  replaced  by 
chlorine,  the  chemical  properties  of  the  original  substance  are  not 
materially  altered.  According  to  Berzelius,  the  three  hydrogen 
atoms  are  positively  charged,  while  the  three  chlorine  atoms  are 
negatively  charged.  That  three  negative  charges  could  be  sub- 
stituted for  three  positive  charges,  in  acetic  acid,  without  pro- 
ducing a  more  marked  change  in  its  properties,  could  not  be 
satisfactorily  explained.  This  criticism  was  for  a  long  time  con- 
sidered to  be  an  insuperable  barrier  to  the  acceptance  of  the 
theory.  Shortly  before  the  close  of  the  nineteenth  century,  J.  J. 
Thomson*  showed  that  this  objection  has  little  or  no  weight. 
When  hydrogen  gas  was  electrolyzed  in  a  vacuum-tube,  and  the 
spectra  at  the  two  electrodes  were  compared,  Thomson  found  them 
to  differ  widely.  From  this  he  concluded  that  the  molecule  of 
hydrogen  gas  is,  in  all  probability,  made  up  of  positively-  and 
negatively-charged  parts,  or  ions.  He  then  extended  his  experi- 
ments to  the  vapors  of  certain  organic  compounds.  In  discussing 
these  experiments  he  says:  — "  In  many  organic  compounds, 
atoms  of  an  electropositive  element,  hydrogen,  are  replaced  by 
atoms  of  an  electronegative  element,  chlorine,  without  altering 
the  type  of  the  compound.  Thus,  for  example,  we  can  replace 
the  four  hydrogen  atoms  in  CH4  by  chlorine  atoms,  getting, 
successively,  the  compounds  CH3C1,  CH2C12,  CHC13,  and  CCU. 
It  seemed  of  interest  to  investigate  what  was  the  nature  of  the 
charge  of  electricity  on  the  chlorine  atoms  in  these  compounds. 
The  point  is  of  some  historical  interest,  as  the  possibility  of  sub- 
stituting an  electronegative  element  in  a  compound  for  an  elec- 
tropositive one  was  one  of  the  chief  objections  against  the  elec- 
trochemical theory  of  Berzelius." 

"  When  the  vapor  of  chloroform  was  placed  in  the  tube,  it  was 
found  that  both  the  hydrogen  and  chlorine  lines  were  bright  on 
the  negative  side  of  the  plate,  while  they  were  absent  from  the 
positive  side,  and  that  any  increase  in  brightness  of  the  hydrogen 
lines  was  accompanied  by  an  increase  in  the  brightness  of  those 
due  to  chlorine.  The  appearance  of  the  hydrogen  and  chlorine 
spectra  on  the  same  side  of  the  plate  was  also  observed  in  methyl- 
ene  chloride  and  in  ethylene  chloride.  Even  when  all  the  hydro- 
gen in  methane  was  replaced  by  chlorine,  as  in  carbon  tetra- 
chloride,  the  chlorine  spectra  still  clung  to  the  negative  side  of 
*  Nature,  52,  451  (1895). 


392  THEORETICAL  CHEMISTRY 

the  plate.  The  same  point  was  tested  with  silicon  tetrachloride 
and  the  chlorine  spectrum  was  brightest  on  the  negative  side  of 
the  plate.  From  these  experiments  it  would  appear,  that  the 
chlorine  atoms  in  the  chlorine  derivatives  of  methane  are  charged 
with  electricity  of  the  same  sign  as  the  hydrogen  atoms  they 
displace." 

Electrical  Units.  In  1827,  Dr.  G.  S.  Ohm  enunciated  his  well- 
known  law  of  electrical  conductance,  viz.  :  —  The  strength  of  the 
electric  current  flowing  in  a  conductor  is  directly  proportional  to  the 
difference  of  potential  between  the  ends  of  the  conductor,  and  inversely 
proportional  to  its  resistance.  If  I  represents  the  strength  of  the 
current,  E  the  difference  of  potential,  and  R  the  resistance,  then 
Ohm's  law  may  be  formulated  as  follows: 

E 


The  unit  of  resistance  is  the  ohm,  that  of  difference  of  potential, 
or  electromotive  force,  the  volt,  and  that  of  current,  the  ampere. 
The  ohm  is  defined  as  the  resistance  of  a  column  of  mercury  106.3 
cm.  long  and  1  sq.  mm.  in  cross  section,  at  0°  C.  The  ampere 
is  defined  as  the  current  which  will  cause  the  deposition  of 
0.001118  gram  of  silver  from  a  solution  of  silver  nitrate  in  1  sec- 
ond. The  volt  may  be  defined  as  the  electromotive  force  neces- 
sary to  drive  a  current  of  1  ampere  through  a  resistance  of  1  ohm. 
The  unit  quantity  of  electricity  is  the  coulomb.  This  amount  of 
electricity  passes  when  a  current  of  a  strength  of  one  ampere  flows 
for  one  second.  One  gram  equivalent  of  any  ion  carries  96,500 
coulombs.  This  quantity  of  electricity  is  known  as  the  faraday 
and  is  commonly  represented  by  the  letter  F. 

As  has  already  been  pointed  out,  any  form  of  energy  may  be 
considered  as  the  product  of  two  factors,  a  capacity  factor  and 
an  intensity  factor.     The  capacity  factor  of  electrical  energy  is 
the  coulomb,  while  the  intensity  factor  is  the  volt,  i.e., 
electrical  energy  =  coulombs  X  volts. 

The  unit  of  electrical  energy,  therefore,  is  the  volt-ampere-second, 
commonly  called  the  watt-second.  One  watt-second  is  the  elec- 
trical work  done  by  a  current  of  1  ampere  flowing  under  an  elec- 
tromotive force  of  1  volt  for  1  second,  and  is  equivalent  to  1  X  107 
C.G.S.  units.  The  thermal  equivalent  of  electrical  energy  may 
be  calculated  from  the  relation 


ELECTRICAL  CONDUCTANCE  393 

electrical  energy  in  absolute  units       , 

r — r-nr : FT — 7 =  neat  equiv.  of  elect,  energy, 

mechanical  equiv.  of  heat  6J' 


or 

1  X  107 


42720  gr.  cm. 


=  0.2389  cal.  =  1  watt-second. 


Faraday's  Law.  If  two  platinum  plates  or  electrodes,  one 
connected  to  the  positive,  and  the  other  to  the  negative  terminal 
of  a,  battery,  are  immersed  in  a  solution  of  sodium  chloride,  it 
will  be  found  that  hydrogen  is  immediately  evolved  at  the  nega- 
tive electrode  and  oxygen  at  the.  positive  electrode.  If  the  salt 
solution  is  previously  colored  with  a  few  drops  of  a  solution  of 
litmus,  it  will  be  observed  that  the  portion  of  the  solution  in  the 
neighborhood  of  the  positive  electrode  will  turn  red,  indicating 
the  formation  of  an  acid,  while  that  in  the  neighborhood  of  the 
negative  electrode  will  turn  blue,  showing  the  formation  of  a 
base.  The  same  changes  will  take  place  whether  the  electrodes 
are  placed  near  together  or  far  apart,  and  furthermore,  the  evolu- 
tion of  gas  and  the  change  in  color  at  the  electrodes  commences 
as  soon  as  the  circuit  is  closed.  The  study  of  these  phenomena 
led  Faraday  *  to  the  conclusion,  that  when  an  electric  current 
traverses  a  solution,  there  occurs  an  actual  transfer  of  matter, 
one  portion  travelling  with  the  current,  and  the  other  portion 
moving  in  the  opposite  direction.  Faraday  termed  these  carriers 
of  the  current,  ions.  He  also  called  the  electrode  connected  to 
the  positive  terminal  of  the  battery,  the  anode,  and  the  electrode 
connected  to  the  negative  terminal,  the  cathode.  The  ions  which 
move  toward  the  anode  he  called  anions,  while  those  which  mi- 
grate toward  the  cathode  he  called  cations.  The  whole  process  he 
termed  electrolysis. 

The  question  of  the  relationship  between  the  amount  of  elec- 
trolysis and  the  quantity  of  electricity  passing  through  a  solution, 
was  first  investigated  by  Faraday.  As  a  result  of  his  experiments 
he  enunciated  the  following  law,  which  is  commonly  known  as  the 
law  of  Faraday: 

(1)  For  the  same  electrolyte,  the  amount  of  electrolysis  is  propor- 
tional to  the  quantity  of  electricity  which  passes. 

(2)  The  amounts  of  substances  liberated  at  the  electrodes,  when  the 
same  quantity  of  electricity  passes  through  solutions  of  different  elec- 

*  Experimental  Researches  (1834). 


394  THEORETICAL  CHEMISTRY 

trolytes,  are  proportional  to  their  chemical  equivalents.  The  chem- 
ical equivalent  of  any  ion  is  defined  as,  the  ratio  of  the  atomic 
weight  of  the  ion  to  its  valence.  If  the  same  quantity  of  elec- 
tricity is  passed  through  solutions  of  hydrochloric  acid,  silver 
nitrate,  cuprous  chloride,  cupric  chloride,  and  auric  chloride,  the 
relative  amounts  of  the  respective  cations  liberated  will  be  as 
follows:  — H*  =  1,  Ag'  =  108,  Cu*  =  63.4,  CiT  =  63.4/2  and 
Au""=  197/3. 

The  electrochemical  equivalent  of  an  element  or  group  of  elements 
is  the  weight  in  grams  which  is  liberated  by  the  passage  of  one 
coulomb  of  electricity.  The  -electrochemical  equivalents  are, 
according  to  Faraday's  law,  proportional  to  the  chemical  equiva- 
lents. 

As  has  been  pointed  out,  the  quantity  of  electricity  necessary 
to  liberate  one  chemical  equivalent  in  grams  is  96,500  coulombs. 
Since  96,500  coulombs  of  electricity  liberate  one  gram  equivalent 
of  hydrogen,  it  follows  that  1.008  -f-  96,500  =  0.0000104463  gram 
of  hydrogen  will  be  liberated  by  1  coulomb.  The  same  quantity 
of  electricity  will  liberate  35.45  X  0.0000104463  =  0.00036749 
gram  of  chlorine,  and  108  X  0.0000104463  -  0.001118  gram  of 
silver.  Or,  in  general,  since  one  coulomb  of  electricity  liberates 
0.0000104463  gram  of  hydrogen,  it  will  cause  the  liberation  of 
0.0000104463  w  grams  of  any  other  element  whose  equivalent 
weight  is  w. 

A  large  amount  of  experimental  work  has  been  done  on  the 
electrolysis  of  solutions  of  silver  nitrate  with  a  view  to  the  pre- 
cise determination  of  the  electrochemical  equivalent  of  silver  and 
the  definition  of  the  ampere.*  The  value  of  the  faraday  has  re- 
cently been  determined  by  Vinal  and  Bates  f  with  an  experimental 
error  not  exceeding  0.01  per  cent.  In  this  investigation,  solutions 
of  silver  nitrate  and  potassium  iodide  were  electrolyzed  in  the 
same  circuit,  and  the  weights  of  silver  and  iodine  liberated  were 
determined.  These  experiments  furnish  an  admirable  confirma- 
tion of  the  laws  of  Faraday,  as  may  be  seen  by  the  following 
table  giving  the  summary  of  the  results  obtained  by  Vinal 
and  Bates. 

*  Bull.  Bureau  of  Standards,  i,  1  (1904);  9,  494  (1912);  10,  425  (1914); 
ii,  220,  555  (1914);  Richards,  Proc.  Am.  Acad.,  37,  415  (1902);  44,  91  (1908); 
Jour.  Am.  Chem.  Soc.,  37,  692  (1915). 

f  Bull.  Bureau  of  Standards,  10,  425  (1914). 


ELECTRICAL  CONDUCTANCE 


395 


DETERMINATION  OF  THE  FARADAY 


Mean 
Silver 
Deposit 

Mean 
Iodine 
Deposit 

Calculated  Coulombs 

Ratio  Silver: 
Iodine 

Electro- 
chemical 
Equivalent 
of  Iodine 

Value  of 
the  Fara- 
day 
(1  =  126.92) 

From  Ag. 
Volta- 
meter 

From  Cell 
and  Res. 

mg 
4105.82 
4104.69 
4099.03 
4397.11 
4105.23 
4123.10 
4104.75 
4184.24 
4100.27 
4105.16 

Weighte 

mg 
4829.59 

4828.62 
4822.24 
5172.73 
4828.51' 
4849.42 
4828.60 
4921.30 
4822.47 
4828.44 

id  mean  ( 

3672.47 
3671.45 
3666.39 
3933.01 
3671.94 
3687.92 
3671.51 
3742.61 
3667.50 
3671.88 

all  observ 

367  i!  53 
3666.55 

367l'.84 

0.850138 
075 
026 
056 
205 
226 
09i 
230 
242 
204 
0.85016 
0.85017 

1.31508 
518 
526 
521 
498 
495 
515 
494 
492 
498 
1.31505 
1.31502 

96  511 
504 
498 
502 
518 
521 
506 
521 
523 
519 
96  514 
96  515 

3671.61 

3667.65 
3671.82 

ations) 

The  law  of  Faraday  is  among  the  very  few  generalizations  to  which 
there  do  not  appear  to  be  any  exceptions.  This  law  has  been 
found  to  hold  with  the  same  exactness  in  both  dilute  and  concen- 
trated solutions,  at  low  and  high  temperatures,*  at  low  and  high 
pressures  and  also  in  all  solvents,  f 

Coulometers.  The  quantity  of  electricity  passing  through  an 
electric  circuit  is  measured  by  means  of  an  electrolytic  cell,  which 
is  so  arranged  as  to  allow  the  amount  of  chemical  change  occurring 
at  an  electrode  surface  to  be  accurately  determined.  Such  a 
cell  is  known  as  a  coulometer.  There  are  three  general  types 
of  coulometer  in  use,  namely,  (1)  weight  coulometers,  in  which 
the  quantity  of  electricity  is  measured  by  the  gain  in  weight 
of  the  cathode,  due  to  the  deposition  of  metal  from  the 
electrolyte;  (2)  volume  coulometers,  in  which  the  volume  of 
electrolytic  gas  liberated  by  the  passage  of  the  current  is 
measured;  and  (3)  titration  coulometers,  in  which  the  change 
in  concentration  of  the  electrolyte  at  one  of  the  electrodes  is 
determined  by  titration  with  a  standard  solution.  Of  these 
three  types,  the  most  accurate  is  the  silver  coulometer  in  which 
silver  is  deposited  from  a  solution  of  silver  nitrate  upon  a 

*  Richards  and  Stull,  Proc.  Am.  Acad.,  38,  409  (1902). 
f  Kahlenberg,  Jour.  Phys.  Chem.,  4,  349  (1900). 


396  THEORETICAL  CHEMISTRY 

weighed  platinum  dish  as  a  cathode.*  For  industrial  purposes 
the  copper  coulometer  is  usually  employed.  This  is  also  a  weight 
coulometer,  and  consists  of  two  sheets  of  pure  copper  as  an  anode 
between  which  is  placed  a  single  weighed  sheet  of  thin  copper  as 
a  cathode,  while  a  solution  of  copper  sulphate  serves  as  the  elec- 
trolyte. 

The  Existence  of  Free  Ions.  When  an  electrolyte  is  decom- 
posed by  the  electric  current,  the  products  of  decomposition  ap- 
pear at  the  electrodes.  The  fact  that  the  liberation  of  the  prod- 
ucts of  decomposition  is  independent  of  the  distance  between 
the  electrodes  caused  considerable  difficulty  in  the  early  history  of 
electrochemistry.  It  was  evident  that  the  two  products  could 
hardly  be  derived  from  the  same  molecule,  but  must  come  from 
two  different  molecules.  Several  theories  were  advanced  to  ac- 
count for  the  experimental  results.  Thus,  in  the  electrolysis  of 
water  it  was  suggested  that  the  two  gases,  hydrogen  and  oxygen, 
were  not  derived  from  the  water,  but  that  electricity  itself  pos- 
sessed an  acid  character.  Grotthussf  was  the  first  to  propose  a 
rational  hypothesis  as  to  the  mechanism  of  electrolysis.  He 
assumed,  that  when  the  electrodes  in  an  electrolytic  cell  are  con- 
nected with  a  source  of  electricity,  the  molecules  of  the  electrolyte 
arrange  themselves  in  straight  lines  between  the  electrodes,  the 
positive  poles  being  directed  toward  the  negative  electrode  and 
the  negative  poles  toward  the  positive  electrode.  When  elec- 
trolysis begins,  the  cation  of  the  molecule  nearest  the  cathode 
is  liberated  at  the  cathode,  and  the  anion  of  the  molecule  nearest 
the  anode  is  liberated  at  the  anode.  The  anion  which  is  left 
free  near  the  cathode  then  combines  with  the  cation  of  the  next  ad- 
joining molecule,  while  the  anion  thus  left  uncombined,  unites  with 
the  cation  of  its  nearest  neighbor.  A  similar  exchange  of  partners 
is  assumed  to  take  place  throughout  the  entire  molecular  chain. 
Under  the  directive  influence  of  the  two  electrodes,  the  newly- 
grouped  molecules  then  rotate,  so  that  all  of  the  positive  poles  face 
the  negative  electrode  and  all  of  the  negative  poles  face  the 
positive  electrode. 

An  inherent  defect  in  this  theory  was  discovered  by  Grove.  { 
From  his  experiments  with  the  oxy-hydrogen  cell,  which  derives 

*  See  Bull.  Bureau  of  Standards,  i,  3  (1904). 
t  Ann.,  de  Chim.  [1],  58,  54  (1806). 
J  Phil.  Mag.,  27,  348  (1845). 


ELECTRICAL  CONDUCTANCE  397 

its  energy  from  the  union  of  hydrogen  and  oxygen,  he  concluded 
that  a  decomposition  of  the  molecules  of  water  is  not  essential 
for  the  evolution  of  these  two  gases,  but  that  the  molecules  must 
be  present  initially  in  a  state  of  partial  decomposition. 

This  suggestion  was  followed  up  by  Clausius.*  He  argued  that 
if  an  expenditure  of  energy  is  necessary  to  decompose  the 
molecules,  electrolysis  should  be  impossible  at  very  low  voltages. 
Experiment  showed,  that  when  silver  nitrate  is  electrolyzed  between 
silver  electrodes,  decomposition  takes  place  at  voltages  which  are 
much  below  the  voltage  corresponding  to  the  energy  of  formation 
of  silver  nitrate.  In  other  words,  it  requires  very  little  energy  to 
decompose  a  salt  which  is  formed  with  the  evolution  of  a  large 
amount  of  energy,  a  result  which  is  in  contradiction  to  the  principle 
of  the  conservation  of  energy.  Clausius  was  thus  forced  to  con- 
clude, "  that  the  supposition  that  the  constituents  of  the  molecule 
of  an  electrolyte  are  firmly  united  and  exist  in  a  fixed  and  orderly 
arrangement  is  wholly  erroneous." 

As  a  result  of  his  investigation  of  the  synthesis  of  ethyl  ether 
from  alcohol  and  sulphuric  acid,  Williamsonf  concluded  "  that  in 
an  aggregate  of  the  molecules  of  every  compound,  a  constant 
interchange  between  the  elements  contained  in  them  is  taking 
place."  In  the  same  paper  he  writes,  "  each  atom  of  hydrogen 
does  not  remain  quietly  attached  all  the  time  to  the  same  atom  of 
chlorine,  but  they  are  continually  exchanging  places  with  one 
another."  This  view  was  accepted  by  Clausius,  although  he 
had  no  means  of  determining  the  extent  to  which  the  electrolyte 
was  broken  down,  or  dissociated  into  free  ions. 

In  1887,  Arrhenius  t  developed  the  views  of  Clausius  by  show- 
ing how  the  degree  of  dissociation  of  the  molecules  of  an  electrolyte 
can  be  deduced  from  measurements  of  the  electrical  conductance 
of  its  solutions,  as  well  as  from  measurements  of  osmotic  pressure 
and  freezing-point  lowering.  The  important  generalization  sum- 
marizing these  conceptions  is  none  other  than  the  theory  of 
electrolytic  dissociation,  to  which  reference  has  already  been 
made  in  an  earlier  chapter  (see  page  212). 

The  Migration  of  the  Ions.  Since  the  passage  of  a  current  of 
electricity  through  a  solution  of  an  electrolyte,  causes  the  dis- 

*  Pogg.  Ann.,  101,  338  (1857). 

f  Lieb.  Ann.,  77,  37  (1851). 

j  Zeit.  phys.  Chem.,  i,  631  (1887). 


398 


THEORETICAL  CHEMISTRY 


charge  of  equivalent  amounts  of  positive  and  negative  ions  at  the 
electrodes,  it  might  be  inferred  that  all  of  the  ions  move  with  the 
same  speed.  That  this  inference  is  incorrect,  was  first  shown  by 
Hittorf  *  as  the  result  of  his  observations  on  the  changes  in  con- 
centration of  the  solution  in  the  neighborhood  of  the  electrodes 
during  electrolysis.  He  showed  that  _  different  .ions  migrate  with 
different  speeds,  and  that  the  faster  moving  ions  carry  a  greater 
proportion  of  the  current  than  the  slower  moving  ions.  The 
effect  of  unequal  ionic  velocities  on  the  concentrations  of  the 
solutions  around  the  electrodes  is  clearly  shown  by  the  accom- 
panying diagram,  (Fig.  106),  due  to  Ostwald.  The  anode  and 


A 

C 

. 

In 

i 

! 

0 

0 

0 

© 

io 

© 

0 

O 

0 

0 

|© 

000 

i 
i 
i 

i 

I 

i 
i 

i 

® 

® 

0 

® 

® 

© 

® 

© 

® 

1® 

®  ©  © 

i 

i 

000 

0 

0 

0 

10 

0 

0 

0 

0 

0 

10 

0 

i 

i 

i 

i 

© 

(• 

® 

i] 

I 

© 

(? 

!© 

®  ®  ®  ®® 

000 

0 

0 

0 

Io 

0 

0 

0 

0 

0 

1 

i© 

0 

! 

i 

j 

i 

V 

* 

_ 

000 

0 

0 

0 

!0 

0 

0 

0 

0 

0 

jo 

0 

1 

Fig.  106 

cathode  in  an  electrolytic  cell  are  represented  by  the  vertical  lines, 
A  and  C,  respectively.  The  cell  is  divided  into  three  compart- 
ments by  means  of  porous  diaphragms,  represented  by  the  ver- 
tical dotted  lines.  The  cations  are  represented  by  dots  (*),  and 
the  anions  by  dashes  (').  Before  the  current  passes  through  the 
cell,  the  concentration  of  the  solution  is  uniform  throughout,  the 
conditions  being  represented  by  I.  Now  let  us  imagine  that  only 
the  anions  move  when  the  circuit  is  closed.  The  conditions, 
when  the  chain  of  anions  has  moved  two  steps  toward  the  anode, 
are  shown  in  II.  Each  ion  which  has  been  deprived  of  a  partner 
is  supposed  to  be  discharged.  It  will  be  observed,  that  although 

*  Pogg.  Ann.,  89,  177;  98,  1;   103,  1;   106,  337,  513  (1853-1859). 


ELECTRICAL  CONDUCTANCE  399 

the  cations  have  not  migrated  toward  the  cathode,  yet  an  equal 
number  of  positive  and  negative  ions  are  discharged,  and  further- 
more that  while  the  concentration  in  the  anode  compartment 
has  not  changed,  the  concentration  in  the  cathode  compart- 
ment has  diminished  to  one-half  its  original  value.  Let  us 
now  suppose  that  both  anions  and  cations  move  with  the 
same  speed,  and  as  before,  let  each  chain  of  ions  move  two  steps 
toward  their  respective  electrodes,  as  indicated  in  III.  It  will  be 
seen,  that  four  positive  and  four  negative  ions  have  been  dis- 
charged, and  that  the  concentration  of  the  electrolyte  in  the 
anode  and  cathode  compartments  has  diminished  to  the  same 
extent.  Finally,  let  us  assume  that  the  ratio  of  the  speed  of  the 
cations  to  that  of  the  anions  is  as  3  :  2.  When  the  cations  have 
moved  three  steps  toward  the  cathode  and  the  anions  have  moved 
two  steps  toward  the  anode,  the  conditions  will  be  as  shown  in 
IV.  It  is  evident  that  five  positive  and  five  negative  ions  have 
been  discharged,  and  that  the  concentration  in  the  cathode  com- 
partment has  diminished  by  two  molecules,  while  the  concentration 
in  the  anode  compartment  has  diminished  by  three  molecules. 
It  will  be  observed,  that  the  change  in  concentration  in  either  of 
the  electrode  compartments  is  proportional  to  the  speed  of  the 
ion  leaving  it.  Thus,  in  II,  the  concentration  in  the  cathode 
compartment  diminishes,  while  that  in  the  anode  compartment 
remains  unchanged,  since  only  the  anion  moves.  In  like  manner, 
the  change  in  concentration  about  the  electrodes  in  III  corre- 
sponds with  the  fact 'that  both  ions  migrate  at  the  same  rate. 
In  IV,  the  ratio  of  the  change  in  concentration  in  the  cathode 
compartment  to  that  in  the  anode  compartment  is  as  2  :  3.  It 
will  be  apparent  from  these  examples,  that  the  relation  between 
the  speeds  of  the  anions  and  the  corresponding  changes  in  concen- 
tration at  the  electrodes  may  be  expressed  by  the  following  pro- 
portion : 

Change  in  concentration  at  anode  due  to  migration   _  speed  of  cation 
Change  in  concentration  at  cathode  due  to  migration  7  speed  of  anion 

If  the  relative  speed  of  the  cations  is  represented  by  u,  and  that 
of  the  anions  by  v,  then  the  total  quantity  of  electricity  trans- 
ported will  be  proportional  to  u  +  v:  of  this  total,  the  fractions 

carried  by  the  anion  and  cation,  respectively,  will  be  na  =  — ^p—  ; 


400 


THEORETICAL  CHEMISTRY 


and   1  —  na  =  nc  = 


The  values  of  these  ratios,  n  and 


n 


n 


U         V 

1  —  n,  are  called  the  transference  numbers  of  the  anion  and  cation 
respectively.  It  is  apparent  from  the  diagram,  that  if  the  elec- 
trolysis is  not  carried  too  far,  the  concentration  of  the  solution 
in  the  intermediate  compartment  will  undergo  no  change.  In 
order  to  determine  transference  numbers,  therefore,  it  is  simply 
necessary  to  remove  portions  of  the  solutions  in  the  immediate 
vicinity  of  the  two  electrodes  and  determine  the  change  in  the 
concentration  of  the  electrolyte  resulting  from  the  passage  of  a 
known  amount  of  electricity.  The  success  of  the  experiment 
depends  upon  keeping  the  concentration  of  the  intermediate 
compartment  unaltered. 

Experimental  Determination  of  Transference  Numbers.  Vari- 
ous forms  of  apparatus  have  been  constructed  for  the  determina- 
tion of  transference  numbers,  among  which 
that  devised  by  the  author,*  and  shown  in 
Fig.  107,  has  proven  very  satisfactory.  It 
consists  of  two  vertical  tubes  of  wide  bore, 
connected  by  a  horizontal  tube  of  the  same 
diameter.  The  electrodes,  of  suitable  metal 
in  the  form  of  wire  are  wound  spirally  around 
the  glass  connecting  tubes.  The  wire  spirals 
are  soldered  to  short  pieces  of  platinum  wire 
which  serve  to  establish  electrical  connection 
with  mercury  in  the  connecting  tubes,  the 
junction  of  the  two  wires  being  covered  with 
a  drop  of  fusion  glass.  The  tubes  by  which 
electrical  connection  is  established  are  held 
in  place  by  means  of  rubber  stoppers 
having  a  groove  cut  on  one  side  to  allow 
for  possible  expansion  of  the  contents  of  the 


Fig.  107 


cell.  The  outlet  tubes  of  the  two  arms  are  provided  with  rubber 
caps  to  protect  them  from  contamination  with  the  water  of 
the  thermostat.  In  carrying  out  a  determination  of  a  trans- 
ference number,  the  cell  is  filled  with  the  solution  of  the 
electrolyte  to  be  studied  and  included  in  a  circuit  containing 
a  variable  resistance,  a  silver  coulometer,  a  milli-ammeter  and  a 
voltmeter.  The  current  is  then  allowed  to  pass  through  the  appa- 
*  Jour.  Am.  Chem.  Soc.,  36,  1640  (1914). 


ELECTRICAL  CONDUCTANCE  401 

ratus  until  a  sufficient  change  in  concentration  has  taken  place 
in  the  anode  and  cathode  compartments,  when  the  circuit  is 
broken  and  the  amount  of  this  change  is  determined  analytically. 
The  portions  of  the  solutions  in  the  two  halves  of  the  apparatus 
are  drawn  out  into  separate  beakers,  and  the  concentration  of  each 
is  determined.  Knowing  the  initial  concentration  of  the  solution 
and  the  final  concentrations  at  the  two  electrodes,  together  with 
the  total  quantity  of  electricity  which  has  passed  through  the 
apparatus  during  the  experiment,  we  have  all  of  the  data  neces- 
sary for  the  calculation  of  the  transference  numbers  of  the  two 
ions. 

The  following  example  will  serve  to  make  the  method  of  calcu- 
lation clear :  —  In  an  experiment  to  determine  the  transference 
numbers  of  the  ions  of  silver  nitrate,  a  solution  containing  0.00739 
gram  of  that  salt  per  gram  of  water  was  prepared.  The  solution 
was  introduced  into  the  migration  apparatus  and,  after  inserting 
silver  electrodes,  a  small  current  was  passed  through  the  appa- 
ratus for  two  hours.  A  silver  coulometer  was  included  in  the  cir- 
cuit, and  0.0780  gram  of  silver  was  deposited  by  the  current. 
This  mass  of  silver  is  equivalent  to  0.000723  gram-equivalent. 
After  the  circuit  was  broken,  the  anode  solution  was  rinsed  out  and 
its  concentration  determined  analytically.  It  was  found  to  con- 
tain 0.2361  gram  of  silver  nitrate  to  23.14  grams  of  water.  This 
amount  of  solution  contained  originally  23.14  X  0.00739  = 
0.1710  gram  of  silver  nitrate.  Thus,  the  amount  of  silver  nitrate 
in  the  anode  compartment  had  increased  by  0.2361  —  0.1710  = 
0.0651  gram  of  silver  nitrate,  or  0.000383  gram-equivalent  of 
silver.  Obviously,  the  increase  in  the  concentration  of  the  nitrate 
ion  must  have  been  the  same.  The  amount  of  silver  dissolved 
from  the  anode  must  have  been  equal  to  that  deposited  in  the 
coulometer,  or  since  0.000723  gram-equivalent  of  silver  was  de- 
posited, and  the  actual  increase  found  was  0.000383  gram-equiva- 
lent, the  difference,  0.000723  -  0.000383  =  0.000340  gram-equi- 
valent, is  the  amount  of  silver  which  migrated  away  from  the 
anode.  At  the  same  time,  0.000383  gram-equivalent  of  nitrate 
ions  migrated  into  the  anode  compartment.  The  ratio  of  the 
speed  of  migration  of  the  silver  ions  to  that  of  the  nitrate  ions  is 
as  0.000340  : 0.000383.  But  0.000723  gram-equivalent  of  silver 
ions  measures  the  total  quantity  of  electricity  transported,  there- 
fore the  transference  numbers  of  the  two  ions  will  be  as  follows : 


402 


THEORETICAL  CHEMISTRY 


Transference  number  of  Ag*  =  nc  =  -    — —  =  0.470, 

0  OOOSSS 
Transference  number  of  N03'  =  na  =  Q'QQQ        =  0.530. 

These  numbers  can  be  checked  by  a  similar  calculation  based  on 
the  change  in  concentration  in  the  cathode  compartment. 

The  following  table  gives  the  transference  numbers  of  the  cat- 
ions of  several  typical  electrolytes,  at  different  concentrations  and 
different  temperatures.  The  transference  number  of  the  corre- 
sponding anions  can  be  found  by  subtracting  the  transference 
numbers  of  the  cations  from  unity. 


TRANSFERENCE  NUMBERS  OF  CATIONS* 


Electrolyte 

Temp. 

Concentration 

0.005 

0.01 

0.02 

0.05 

0.10 

0.20 

NaCl  

0° 

18 
30 
0 
18 
30 
18 
18 
18 
30 
0 
18 
20 
0 
25 
20 
20 
18 
18 

0.387 

0.396 
0.404 
0.493 
0.496 
0.498 

o!isi 

0.847 
0.832 
0.839 
0.439 

0^440 
6!388 

0.387 
0.396 
0.404 
0.493 
0.496 
0.498 
0.332 
0.492 
0.471 
0.481 
0.846 
0.832 
0.840 
0.437 

0.432 
0^385 

0.387 
0.396 
0.404 
0.493 
0.496 
0.498 
0.328 
0.492 
0.471 
0.481 
0.844 
0.833 
0.841 
0.432 

0^424 
0.822 
0.381 
0.375 

0.386 
0.395 
0.404 
0.493 
0.496 
0.498 
0.320 
0.492 
0.471 
0.481 
0.839 
0.833 
0.844 

0.385 
0.393 
0.403 
0.492 
0.495 
0.497 
0.313 

0^471 
0.481 
0.834 
0.834 

6  '.491 
0.494 
0.496 
0.304 

6!  481 
6!  837 

KC1 

LiCl 

NH4C1    

AgNO3     

HC1  

HNO3.. 

BaCl2  

CaCl2 

0.438 
0.413 
0.822 
0.373 
0.375 

0.427 
0.404 
0.822 

6^373 

0.415 
0.395 
0.820 

o!s<5i 

H2SO4 

MgSO4 

CuSO4 

*  Noyes  and  Falk,  Jour.  Am.  Chem.  Soc.,  33,  1454  (1911). 
* 

It  is  apparent  from  the  table  that  the  transference  numbers  are 
not  entirely  independent  of  the  concentration,  and  also  that  they 
vary  slightly  with  the  temperature.  In  fact,  with  rising  tempera- 
ture, the  transference  numbers  of  all  ions  tend  to  become  equal, 
the  limiting  value  beng  0.5. 


ELECTRICAL  CONDUCTANCE 


403 


Transference  Numbers  and  Ionic  Hydration.  In  the  method 
just  outlined  for  the  determination  of  transference  numbers,  it  was 
assumed  that  the  solvent  remains  stationary  during  the  passage  of 
the  current,  and  that  the  change  in  concentration,  which  occurs 
in  the  vicinity  of  the  electrode,  is  due  to  the  difference  in  the  rela- 
tive speeds  with  which  the  ions  move  through  the  solution. 

That  this  assumption  is  not  correct  has  been  proven  by  Wash- 
burn  and  Millard,*  who  have  demonstrated  experimentally  that 
the  ions  are  hydrated  and  carry  a  portion  of  the  solvent  with  them 
from  one  electrode  to  the  other.  While  the  amount  of  solvent 
thus  transferred,  in  dilute  solutions,  is  exceedingly  small,  it  be- 
comes sufficiently  large,  in  concentrated  solutions,  to  affect  the 
values  of  the  transference  numbers.  If  an  inert  substance,  such 
as  sucrose,  be  added  to  the  solution  in  a  transference  experiment, 
it  will  remain  stationary  during  the  passage  of  the  current,  and 
may  be  used  as  a  reference  substance  to  determine  the  amount  of 
solvent  which  has  been  transported  by  the  ions.  In  this  manner, 
Washburn  and  Millard  determined  the  so-called  "  true  "  trans- 
ference numbers  given  in  the  accompanying  table. 

TRUE  TRANSFERENCE  NUMBERS  AT  25° 


Electrolyte 
Concentration 
1.3  Molar 

nw/ncf 

nw 

nc' 

"oo 

nc 

HC1 

0  28 

0.24 

0  844 

0  833 

0  820 

CsCl  .  . 

0.67 

0  33 

0  491 

0  491 

0  485 

KC1  ... 

0.3 

0  60 

0  495 

0  495 

0  482 

NaCl  
LiCl  

2.0 

4.7 

0.76 
1.5 

0.383 
0  304 

0.396 
0  330 

0.366 
0  278     • 

HW  =  number  of  mols  of  water  transferred  from  anode  to  cathode  per 

faraday, 

nc'  =  true  transference  number  of  cation, 
Wo,  =  transference  number  at  infinite  dilution, 
nc   =  Hittorf  transference  number. 

In  dilute  solutions,  the  difference  between  Hittorf  and  true  trans- 
ference numbers  is  so  small  as  to  be  negligible.  If  nw  denotes 
the  number  of  mols  of  water  transferred  from  the  anode  to  the 
cathode  per  faraday  of  electricity,  and  NS/NW  is  the  ratio  of  the 

*  Jour.  Am.  Chem.  Soc.,  37,  694  (1915). 


404  THEROETICAL  CHEMISTRY 

number  of  mols  of  the  solute  to  the  number  of  mols  of  water  in 
the  original  solution,  the  relation  between  the  Hittorf  transfer- 
ence number  of  the  anion,  na,  and  the  true  transference  number 
of  the  anion,  na',  will  be  given  by  the  equation, 

N 
na'  =  na  -  nwlr~-  (2) 

MW 

Specific,  Molar  and  Equivalent  Conductance.  As  is  well 
known,  the  resistance  of  a  metallic  conductor  is  directly  propor- 
tional to  its  length  and  inversely  proportional  to  its  area  of  cross- 
section.  Similarly,  the  resistance  of  an  electrolyte  is  propor- 
tional to  the  length,  and  inversely  proportional  to  the  cross-sec- 
tion of  the  column  of  solution  between  the  two  electrodes.  The 
specific  resistance  of  an  electrolyte  may  be  denned  as  the  resist- 
ance in  ohms  of  a  column  of  solutiomme  centimeter  long  and  one 

"""'" " ~ 

square  centimeter  in  cross-section.  Specific  conductance  is  the 
reciprocal  of  specific  resistance.  Since  the  conductance  of  a  solu- 
tion is  almost  wholly  dependent  upon  the  amount  of  sojute^resent, 
it  is  more  convenient  to  express  conductance  in  terms  of  the  molar, 
or  equivalent  concentration.  The  molar  conductance  //,  is  the 
conductance,  in  reciprocal  ohms,  of  a  solution  containing  one  mol 
of  solute  when  placed  between  electrodes  which  are  exactly  one 
centimeter  apart.  The  equivalent  conductance  A,  is  the  conduct- 
ance, in  reciprocal  ohms,  of  a  solution  containing  one  gram-equi- 
valent of  solute  when  placed  between  electrodes  which  are  one 
centimeter  apart.  If  L  denotes  the  specific  conductance  of  a 
solution,  and  Vm  the  volume  in  cubic  centimeters  which  contains 
one  mol  of  solute,  then 

M  =  LVm, 
and  in  like  manner, 

A  =  LVe,  (4) 

where  Ve  is  the  volume  of  solution  in  cubic  centimeters  which 
contains  one  gram-equivalent  of  solute.  If  C  denotes  the  con- 
centration of  a  solution  of  an  electrolyte  in  gram-equivalents  per 
liter,  equation  (4)  becomes, 

1000  L 
A  =    -£-  (5) 

The  following  table  gives  the  specific  and  equivalent  conduct- 
ance of  solutions  of  sodium  chloride  at  18°  C. : 


ELECTRICAL  CONDUCTANCE  405 

CONDUCTANCE  OF  SOLUTIONS  OF  SODIUM  CHLORIDE 


Concentration 

Dilution 

Sp.  Cond. 

Molar  Cond. 

1 

1,000 

0.0744 

74.4 

0.1 

10,000 

0.00925 

92.5 

0.01 

100,000 

0.001028 

102.8 

0.001 

1,000,000 

0.0001078 

107.8 

0.0001 

10,000,000 

0.00001097 

109.7 

It  will  be  observed  that  the  equivalent  conductance  increases 
with  the  dilution  up  to  a  certain  point,  beyond  which  it  remains 
nearly  constant.  That  the  equivalent  conductance  should  change 
but  little  will  become  apparent  from  the  following  considerations. 
Imagine  a  rectangular  cell  of  indefinite  height  and  having  a  cross- 
sectional  area  of  one  square  centimeter,  and  further  assume  that 
two  opposite  walls  can  function  as  electrodes.  Let  1000  cc.  of  a 
solution,  containing  one  equivalent  of  solute,  be  introduced  into 
the  cell,  and  let  its  conductance  be  determined.  Now  let  the 
solution  be  diluted  to  2000  cc.,  and  the  conductance  of  the  diluted 
solution  be  measured.  While  the  specific  conductance  of  the 
diluted  solution  is  reduced  to  one-half  of  its  original  value,  yet, 
since  the  solution  stands  at  twice  the  original  height  in  the  cell, 
the  electrode  surface  in  contact  with  the  solution  is  doubled,  and 
the  total  conductance  due  to  one  mol  of  solute  remains  unchanged^ 
This,  of  course,  is  only  the  case  with  completely  ionized  solutes. 

Determination  of  Electrical  Conductance.  The  determination 
of  the  electrical  conductance  of  a  solution  resolves  itself  into 
the  determination  of  its  resistance,  by  a  simple  modification 
of  the  familiar  Wheatstone-bridge  method.  The  arrangement  of 
the  apparatus  for  this  method,  devised  by  Kohlrausch,*  is  repre- 
sented diagrammatically  in  Fig.  108,  where  ab  is  the  bridge  wire, 
B  is  a  resistance  box,  and  C  is  a  cell  containing  the  solution  whose 
resistance  is  to  be  measured.  The  points  d  and  c  are  connected 
to  a  small  induction  coil,  7,  which  gives  an  alternating  current. 
This  is  necessary  in  order  to  prevent  polarization  which  would 
occur  if  a  direct  current  were  used.  The  use  of  the  alternating 
current  necessitates  the  substitution  of  a  telephone,  T,  for  a 
galvanometer,  usually  employed  in  measuring  resistance.  The 


Wied.  Ann.,  6,  145  (1879);  u,  653  (1880);  26,  161  (1885). 


406 


THEORETICAL   CHEMISTRY 


positions  of  the  induction  coil  and  telephone  are  sometimes  inter- 
changed, but  the  arrangement  shown  in  the  diagram  is  to  be  pre- 
ferred, since  it  insures  a  high  electromotive  force  where  the  sliding 
contact,  c,  touches  the  wire,  this  being  the  most  uncertain  connec- 
tion in  the  entire  arrangement.  A  small  accumulator,  A,  serves 


Fig.  108 

to  operate  the  induction  coil.  In  making  a  measurement,  the 
coil  is  connected  with  the  accumulator,  and  the  vibrator  is  so  ad- 
justed that  a  high  mosquito-like  tone  is  emitted:  then  the  sliding 
contact,  c,  is  moved  along  the  wire,  ab,  until  the 
sound  in  the  telephone  reaches  a  minimum,  when 
the  position  of  the  point  of  contact  is  read  on  a 
millimeter  scale  placed  below  the  bridge- wire.  Ac- 
cording to  the  principle  of  the  Wheatstone  bridge, 
it  follows,  that 


C 
B 


be 
ac 


Since  the  resistance,  B,  and  the  lengths,  be  and  ac, 
are  known,  the  resistance,  C,  can  be  calculated. 
Various  types  of  conductance  cells  are  in  use,  depending  upon 
whether  the  solution  has  a  high,  or  a  low  resistance.  The 
form  shown  in  Fig.  109  is  widely  used.  The  two  electrodes 
are  made  of  platinum  foil,  connection  with  the  mercury  in  the 
two  glass  tubes,  tt,  being  established  by  means  of  two  pieces 
of  stout  platinum  wire  sealed  through  the  ends  of  these 


ELECTRICAL  CONDUCTANCE  407 

tubes.  The  tubes,  tt,  are  fastened  into  a  tight-fitting  vulcanite 
cover  so  that  the  electrodes  may  be  removed,  rinsed  and  dried, 
without  altering  their  relative  positions.  Before  the  cell  is  used, 
the  electrodes  must  be  coated,  electrolytically,  with  platinum  black. 
It  is  not  necessary  to  know  the  area  of  the  electrodes,  or  the  dis- 
tance between  them,  since  it  is  possible  to  determine  a  factor, 
termed  the  resistance  capacity,  by  means  of  which  the  results  ob- 
tained with  the  cell  can  be  transformed  into  reciprocal  ohms.  To 
this  end,  the  specific  conductances  of  a  number  of  standard  solu- 
tions have  been  carefully  determined  by  Kohlrausch;  thus,  for 
a  0.02  molar  solution  of  potassium  chloride  he  found  the  following 

values: 

L18o  =  0.002397        and        L25°  =  0.002768, 
or 

Ai8o  =  119.85        and        A25o  =  138.4. 

Let  the  resistance  of  the  cell,  when  filled  with  0.02  molar  potassium 
chloride,  be  C,  then,  according  to  the  principle  of  the  Wheatstone 
bridge,  we  have 

r      «   bc 

C    =    X5  •  —  ; 

ac 

or,  denoting  the  conductance  of  the  solution  by  S,  we  obtain, 

„       1          ac 

O    =   77  = 


C      B-bc 

Since  the  specific  conductance,  L,  must  be  proportional  to  the 
observed  conductance,  we  have 

L  =  KS  =  K  ~ , 
B  •  bc 

where  K  is  the  resistance  capacity  of  the  cell.     If  the  measure- 
ment is  made  atJL8°  C., 

0.002397  B  •  bc 


K 


ac 


Having  determined  the  resistance  capacity  of  the  cell,  we  may 
proceed  to  determine  the  conductance  of  any  solution.  For 
example,  suppose  that  when  the  resistance  in  the  box  is  B',  the 
point  of  balance  on  the  bridge-wire  is  at  c',  then  the  specific  con- 
ductance of  the  solution  will  be 


408 


THEORETICAL  CHEMISTRY 


If  L  is  divided  by  the  concentration  in  gram-equivalents  per  liter 
of  the  solution, 

1000  L 
A=     — 

Relative  Conductances  of  Different  Substances.  The  study 
of  the  electrical  conductance  of  various  solutes  in  aqueous  solu- 
tion, reveals  the  fact  that  electrolytes  differ  greatly  in  their  con- 
ducting power.  They  may  be  roughly  divided  into  two  classes :  — 
those  with  high  conducting  power,  such  as  strong  acids,  strong 
bases,  and  salts;  and  those  with  low  conducting  power,  such  as 
ammonia,  and  most  of  the  organic  acids  and  bases.  The  follow- 
ing table  gives  the  equivalent  conductances  of  several  typical 
electrolytes,  from  normal  down  to  so-called  "  zero  "  concentration. 

EQUIVALENT  CONDUCTANCES  OF  TYPICAL  ELECTROLYTES* 

Temp.  18° 


Normal 
Cone. 

NaCl 

KCl 

HCl 

NaNOs 

KNOs 

AgNOs 

HNOs 

0.0000 

108.9 

130.0 

380.0 

105.2 

126.3 

115.8 

376.5 

0.0001 

108.0 

129.0 

378.1 

104.5 

125.4 

115.0 

0.0005 

107.1 

128.0 

377.0 

103.5 

124.3 

113.9 

373  '.9 

0.0010 

106.4 

127.3 

375.9 

102.8 

123.6 

113.1 

372.9 

0.0020 

105.5 

126.2 

375.3 

101.8 

122.5 

112.1 

371.4 

0.0050 

103.7 

124.3 

372.6 

99.97 

120.4 

110.0 

371.0 

0.0100 

101.9 

122  .4 

369.3 

98.07 

118.1 

107.8 

365.0 

-OJ)200 

99.6 

019J, 

365.5 

95.57 

115.1 

105.1 

364.0 

0.0500 

95.7 

115.7 

358.4 

91.35 

109.8 

99.5 

353.7 

0.1000 

92.0 

112.0 

351.4 

87.16 

104.7 

94.3' 

346.4 

0.2000 

87.7 

107.9 

342.0 

82.21 

98.7 

340.0 

0.5000 

80.9 

102.4 

327.0 

73.99 

89.2 

'77  '.5 

324.0 

1.0000 

74.3 

98.2 

301.0 

65.81 

80.4 

67.6 

310.0 

Normal 
Cone. 

Na2SO4 

H2SO4t 

CaCh 

NaOHf 

KOHf 

NH4OH 

CHsCOOH 

0.0000 

111.9 

117.4 

239.00 

350.00 

0.0001 

109.3 

115.2 

66.00 

107.00 

0.0005 

106.7 

113.3 

38.00 

57.00 

0.0010 

105.1 

36l'.6 

112.0 

234.0 

28.00 

41.00 

0.0020 

103.5 

351.0 

110.1 

204.5 

233.0 

20.60 

30.20 

0.0050 

99.8 

330.0 

106.7 

230.0 

13.20 

20.00 

0.0100 

95.7 

308.0 

103.4 

203.4 

228.0 

9.60 

14.30 

0.0200 

286.0 

99.4 

225.0 

7.10 

10.40 

0.0500 

'83^6 

253.0 

93.3 

199  '.0 

219.0 

4.60 

6.48 

0.1000 

77.1 

225.0 

88.2 

195.4 

213.0 

3.30 

4.60 

0.2000 

70.0 

214.0 

82.8 

206.0 

2.30 

3.24 

0.5000 

205.0 

74.9 

iri'.i 

197.0 

1.35 

2.01 

1.0000 

198.0 

67.6 

157.0 

184.0 

0.89 

1.32 

*  Noyes  and  Falk,  Jour.  Am.  Chem.  Soc.,  34,  461  (1912). 
t  Landolt  and  Bornstein  Tabellen. 


ELECTRICAL  CONDUCTANCE  409 

In  every  case  it  will  be  observed,  that  the  equivalent  conductance 
increases  with  the  dilution  of  the  solution  until  a  limiting  value, 
AO,  is  reached.  It  is  important  to  bear  in  mind,  that  although 
AO  is  frequently  referred  to  as  the  conductance  at  infinite  dilu- 
tion, it  is  not  identical  with  the  conductance  of  the  pure  sol- 
vent. 

In  the  case  of  so-called  "  strong  electrolytes,"  the  value  of  the 
limiting  conductance,  A0,  is  obtained  by  extrapolating  the  values 
of  the  conductance  at  finite  concentrations.  Although  the  accu- 
racy of  extrapolated  data  is  always  more  or  less  uncertain,  the 
error  in  the  extrapolated  values  of  A0,  for  strong  electrolytes,  is 
known  to  be  less  than  one  per  cent,  in  the  majority  of  cases.  To 
minimize  the  error  in  extrapolation,  Noyes  and  Falk  *  plotted  the 
reciprocal  of  the  equivalent  conductance,  I/A,  against  (CA)m  and 
then,  by  trial,  determined  the  value  of  m  which  would  give 
a  straight  line.  On  extending  this  line  until  it  cuts  the  axis  at  a 
point  corresponding  to  (CA)m  =  0,  the  limiting  value  of  I/A  was 
found,  and  from  this  the  value  of  the  limiting  conductance,  A0, 
could  easily  be  calculated.  The  value  of  m  was  found  to  be  ap- 
proximately 0.5. 

The  limiting  conductance  of  weak  electrolytes  cannot  be  deter- 
mined by  the  method  just  outlined,  owing  to  the  fact  that,  at 
concentrations  below  0.0001  molar,  it  is  very  difficult  to  make 
satisfactory  measurements  of  conductance,  and  consequently  the 
method  of  extrapolation  cannot  be  employed.  In  order  to  deter- 
mine the  value  of  A0,  for  weak  electrolytes,  it  is  necessary  to  know 
the  values  of  the  conductance  of  the  individual  ions,  as  determined 
by  the  law  of  Kohlrausch. 

The  Law  of  Kohlrausch.  The  electrical  conductance  of  solu- 
tions was  systematically  investigated  by  Kohlrausch  who  showed 
that  the  limiting  value  of  the  equivalent  conductance,  which  may 
be  represented  by  A0,  is  different  for  different  electrolytes,  and 
may  be  considered  as  the  sum  of  two  independent  factors,  one  of 
which  refers  to  the  cation  and  the  other  to  the  anion.  This  experi- 
mental result  is  ^ 


The  limiting  value  of  the  equivalent  conductance  is  reached 

when  the  molecules  are  completely  broken  down  into  ions,  and, 

under  these  conditions,  the  whole  of  the  electrolyte  participates 

in  conducting  the  current.     The  accompanying  table,  giving  the 

*  Jour.  Am.  Chem.  Soc.,  34,  454  (1912). 


410 


THEORETICAL  CHEMISTRY 


equivalent   conductances  at   infinite   dilution   of   several  binary 
electrolytes,  illustrates  the  truth  of  the  law  of  Kohlrausch. 

EQUIVALENT  CONDUCTANCES  AT  INFINITE  DILUTION 

Temp.  18° 


K 

Na 

Li 

NH~4 

H 

Ag 

Cl 

130 

108 

98 

130 

378 

NO*.. 

126 

105 

375 

116 

OH  

238 

217 

C1O3  

119 

109 

C2H3O2  ... 

99 

78 

89 

The  differences  between  two  corresponding  sets  of  numbers  in 
the  same  vertical  column,  or  between  any  two  corresponding  sets 
of  numbers  in  the  same  horizontal  row,  will  be  found  to  be 
nearly  equal.  This  could  only  occur  when  the  limiting  conduct- 
ance represents  the  sum  of  two  entirely  independent  quantities.  • 

Each  ion  invariably  carries  the  same  charge  of  electricity,  and 
moves  with  its  own  velocity,  quite  independent  of  the  nature  of 
its  companion  ion.  Therefore,  at  infinite  dilution,  we  have 


Ao   =   lc  + 


(6) 


in  which  lc  and  la  are  the  equivalent  conductances  of  the  ions  of 
the  electrolyte  at  infinite  dilution.     From  this  it  follows,  that 


and 

or 
and 


V 


u 

la  = 
lc  = 


(7) 

(8) 

(9) 
(10) 


Thus,  the  equivalent  conductance  of  silver  nitrate  at  infinite  dilu- 
tion, at  18°,  is  115.5,  while  na  =  0.518  and  nc  =  0.482;  therefore, 


and 


la  =  0.518  X  115.5  =  59.8, 
lc  =  0.482  X  115.5  =  55.7; 


ELECTRICAL  CONDUCTANCE 


411 


or,  one  gram-equivalent  of  N(V  ions  possesses  a  conductance  of 
59.8,  when  placed  between  electrodes  one  centimeter  apart,  and 
large  enough  to  contain  between  them  the  entire  volume  of  solu- 
tion in  which  the  N(V  ions  exist.  Under  the  same  conditions, 
one  gram-equivalent  of  Ag*  ions  has  a  conductance  equal  to 
55.7. 

The  values  of  the  ionic  conductances,  at  infinite  dilution,  remain 
constant  in  all  solutions  in  the  same  solvent  at  the  same  temper- 
ature, so  that  it  is  possible  to  calculate  the  equivalent  conductance 
for  any  substance  at  infinite  dilution. 

In  the  subjoined  table  are  given  the  ionic  conductances  of 
various  ions  at  18°  and  infinite  dilution,  together  with  their  tem- 
perature coefficients. 


IONIC  CONDUCTANCES  AT  INFINITE  DILUTION 

Temp.  18° 


-- 


Ion 

Ic 

Temp. 
Coeff. 

Ion 

Ic 

Temp. 
Coeff. 

Li' 

33  07 

0.0265 

C1O3'.. 

54.95 

0.0215 

Na* 

43  23 

0  0244 

IO3' 

33.83 

0.0234 

K* 

64.30 

0.0217 

NO3'  

61.71 

0.0205 

Rb* 

67  6 

0  0214 

H*               .... 

313.0 

Cs' 

67  56 

0  0212 

OH'           

174.0 

NH4* 

64  5 

0  0222 

A  Zn"  

45.6 

0.0251 

Tl*  

65.5 

0.0215 

iMg"  

45.0 

0.0256 

As' 

53  9 

0  0229 

i  Ba"  

55.0 

0.0238 

cr 

65  34 

0  0216 

i  Pb"  

61.0 

0.0243 

Br' 

67.41 

0  0215 

i  SO4"  

68.0 

0.0227 

r.  .. 

66.35 

0.0213 

i  CO3"  

70.0 

0.0270 

In  the  case  of  weak  electrolytes,  the  value  of  A0  cannot  be  deter- 
mined directly  from  conductance  measurements,  since  the  solu- 
tion becomes  so  dilute  before  the  limiting  value  of  the  conduct- 
ance is  reached  that  accurate  measurements  of  the  specific  con- 
ductance are  impossible.  However,  the  law  of  Kohlrausch  en- 
ables us  to  overcome  this  difficulty.  Thus,  the  value  of  A0  for 
acetic  acid  must  be  equal  to  the  sum  of  the  conductances  of  the 
H*  and  CH3COO'  ions.  The  conductance  of  the  H'  ion  at  18°  is 
313,  according  to  the  preceding  table,  while  the  value  of  the  con- 
ductance of  sodium  acetate  at  infinite  dilution  is  78.1  at  18°. 
Since  the  ionic  conductance  of  the  Na*  ion  is  43.23  at  18°,  it  fol- 


412 


THEORETICAL  CHEMISTRY 


lows  that  the  conductance  of  the  CH3COO'  ion  must  be,  78.1  — 
43.23  =  34.87.     Therefore,  for  acetic  acid,  we  have 

Ao  =  lc  +  la  =  313  +  34.87  =  347.87  at  18°. 

Bredig  *  has  shown  that  the  ionic  conductance  of  elementary 
ions  is  a  periodic  function  of  the  atomic  weight.  When  the  ionic 
conductances  are  plotted  as  ordinates  against  the  atomic  weights 
as  abscissae,  the  curve,  shown  in  Fig.  110,  is  obtained.  A  glance  at 
the  curve  shows  the  periodic  nature  of  the  relation. 

Conductance  and  lonization.  We  have  already  seen  that 
solutions  of  strong  acids,  strong  bases  and  salts  exert  abnormally- 
great  osmotic  pressures.  According  to  the  molecular  theory,  this 
abnormal  osmotic  activity  has  been  ascribed  to  the  presence  in 
the  solutions  of  a  greater  number  of  dissolved  particles  than  would 
be  anticipated  from  the  simple  molecular  formulas  of  the  solutes. 
The  ratio  of  the  observed  to  the  theoretical  osmotic  pressure  was 
represented,  according  to  van't  Hoff,  by  the  factor  "  i" 


80- 


60- 


40- 


20- 


40 


Atomic  Weight 
Fig.  110 


120 


160 


200 


That  there  is  an  intimate  connection  between  electrical  con- 
ductance and  abnormal  osmotic  activity,  is  shown  by  the  fact  dis- 
covered by  Arrhenius  in  1887,  that  only  those  solutions  conduct 
the  electric  current  which  exert  abnormally-high  osmotic  pres- 
sures. It  had  already  been  pointed  out  by  Kohlrausch,  that  the 
*  Zeit.  phys.  Chem.,  13,  242  (1894). 


ELECTRICAL  CONDUCTANCE  413 

equivalent  conductance  of  a  solution  increases  initially  with  the 
dilution,  and  then,  ultimately,  becomes  constant.  Arrhenius  ex- 
plained this  behavior  by  assuming  that  the  molecules  of  the 
solute  undergo  dissociation  into  ions.  The  degree  of  dissociation 
increases  with  the  dilution  until  finally,  when  the  equivalent 
conductance  has  reached  its  maximum  value,  the  molecules  of 
solute  become  completely  broken  down  into  ions.  According  to 
the  theory  of  electrolytic  dissociation,  the  conductance  of  a  solu- 
tion is  dependent  upon  the  number  of  ions  present  in  the  solution, 
upon  the  magnitude  of  their  charges,  and  upon  their  velocities. 
Since  the  electric  charges  carried  by  equivalent  amounts  of  the 
ions  of  different  electrolytes  are  equal,  and  since  the  velocities  of 
the  ions  for  the  same  electrolyte  are  practically  independent  of  the 
dilution  of  the  solution,  it  follows  that  the  increase  in  equivalent 
conductance  with  dilution  must  depend  almost  wholly  upon  the 
increase  in  the  number  of  ions  present. 

The  equivalent  conductance  of  an  electrolyte  at  infinite  dilution 
has  been  shown  to  be 

Ao  =  le  +  Z«, 

and,  therefore,  the  equivalent  conductance  at  any  dilution  v,  must 

be 

A  =  a  (lc  +  U, 

where  a  is  the  degree  of  dissociation  of  the  electrolyte.  Dividing 
the  second  equation  by  the  first,  we  obtain, 

-=£•  (ID 

Ao 

This  equation  enables  us  to  calculate  the  degree  of  ionization  of 
an  electrolyte  at  any  concentration,  provided  we  know  the  con- 
ductance of  its  solution  at  the  given  concentration,  and  also  its 
conductance,  at  infinite  dilution.  It  should  be  borne  in  mind, 
that  equation  (11)  was  derived  on  the  assumption,  that  in  any 
solution  the  velocities  of  the  anions  and  cations  are  the  same  as  at 
infinite  dilution.  Since  the  solutions  of  many  electrolytes  are 
known  to  possess  high  viscosities,  it  is  more  than  probable  that 
in  these  solutions  the  ions  encounter  very  appreciable  resistance, 
in  consequence  of  which  their  velocities  become  more  and  more 
reduced  as  the  concentration  is  increased.  If  we  assume  that 
Stokes'  law  is  applicable  to  the  motion  of  an  ion  through  a  solu- 
tion, that  is,  that  the  equivalent  conductance  of  an  ion  is  inversely 


414 


THEORETICAL  CHEMISTRY 


proportional  to  the  viscosity  of  the  medium  through  which  it 
moves,  it  follows,  that  equation  (11)  may  be  written  in  the  form, 


A    77 
a  =  —  •  — 

AQ     170 


(12) 


where  rj  and  170  are  the  viscosities  of  the  solution  and  solvent,  re- 
spectively. In  some  cases  it  has  been  found,  that  the  values 
calculated  by  means  of  equation  (12)  are  in  better  agreement  with 
the  results  obtained  from  freezing-point  measurements  than  are 
the  values  calculated  by  means  of  equation  (11);  in  other  cases, 
the  latter  equation  is  found  to  give  the  better  results.  At  the 
present  time  it  may  be  fairly  assumed  that  the  simpler  equation 
gives  a  correct  measure  of  the  degree  of  ionization  of  all  uni- 
univalent  electrolytes  in  dilute  solution. 

Ionization  of  Salts,  Acids  and  Bases.  The  values  of  the  con- 
ductance ratio,  A/AO,  for  salts  of  the  same  general  type,  are  prac- 
tically identical  at  the  same  equivalent  concentrations.  This  is 
clearly  shown  by  the  accompanying  table,  compiled  by  Randall,* 
giving  the  ionization  values  of  typical  salts  of  various  valence 
types  at  18°. 

VALUES    OF  THE  CONDUCTANCE  RATIO,   A/A0,   FOR 

TYPICAL  SALTS 

Temp.  18° 


Salt 

Concentration  (gram-mols  per  liter) 

0.0001 

0.0002 

0.0005 

0.001 

0.002 

0.005 

0.01 

0.05 

0.10 

KCL. 

99.2 

99.1 
98.4 
98.4 

99.0 

98.7 
97.8 
97.8 

98.5 
97.9 
96.7 
96.7 
96.4 
96.0 

97.9 
97.0 
95.5 
95.3 
94.8 
94.3 

86^2 

97.1 

95.8 
93.9 
93.4 
92.4 
91.7 
85.9 
80.4 

95.6 
93.6 
91.0 

89.8 
88.2 
86.5 

70^9 

94.1 
91.2 

88.3 
86.1 
83.7 
80.8 
71.2 
62.9 

88.9 
83.4 
80.3 
74.4 
69.4 
62.7 
59.1 

86.0 
78.9 
76.5 
67.9 
62.5 

LiIO3 

MgCl2  
Ba(NO)3 

T12SO4.  . 

PbCl2.. 

K4Fe(CN)6... 
CuSO4  

96.1 

94.4 

90.5 

In  the  case  of  uni-univalent  and  bi-bivalent  salts,  the  conductance 
ratio  undoubtedly  affords  a  true  measure  of  the  degree  of  ioniza- 
tion, because  salts  belonging  to  these  types  dissociate  directly 
*  Jour.  Am.  Chem.  Soc.,  38,  788  (1916). 


ELECTRICAL  CONDUCTANCE 


415 


into  the  ions  corresponding  to  the  equivalent  conductance  at 
infinite  dilution.  As  has  been  pointed  out  in  a  previous  chapter, 
salts  of  higher  types  dissociate  in  stages,  giving  rise  to  interme- 
diate ions.  For  example,  a  bi-univalent  salt,  such  as  CaCl2,  disso- 
ciates in  the  following  manner: 

(a)  CaCl2  ->  Cad'  +  Cl', 

(b)  CaCl"  *»  Ca"  +  Cl'. 

At  infinite  dilution,  when  ionization  is  complete,  only  two  species 
of  ions  will  be  present,  but  at  any  finite  concentration,  there  will 
always  be  three  ionic  species  present  in  unknown  relative  amounts. 
Obviously,  equation  (11)  cannot  be  employed  to  calculate  the 
degree  of  ionization  when  more  than  two  species  of  ions  are  pres- 
ent. An  indirect  method  for  the  determination  of  the  degree 
of  ionization  of  certain  salts  which  yield  intermediate  ions  has 
been  developed  by  Harkins.* 

With  very  few  exceptions,  all  salts  are  highly  ionized  in  dilute 
solution.  Among  the  most  noteworthy  exceptions  are  the 
mercury  halides  and  cadmium  chloride;  in  0.1  molar  solution,  the 
degree  of  ionization  of  the  former  is  only  0.1  per  cent  while  that 
of  the  latter  is  45  per  cent.  No  satisfactory  explanation  has  been 
advanced  to  account  for  these  anomalies. 

While,  in  equivalent  concentrations,  salts  of  the  same  ionic 
type  are  ionized  to  practically  the  same  extent,  the  degree  of  ion- 
ization of  acids  and  bases  is  found  to  vary  in  a  most  surprising 
manner.  This  is  apparent  from  the  following  table,  in  which  the 
degree  of  ionization  of  several  typical  acids  and  bases  is  recorded. 

DEGREE  OF  IONIZATION  OF  ACIDS  AND  BASES  AT  18° 


A 

Concentration 

AO 

HC1 

HNOs 

HC!H,0, 

KOH 

NH*OH 

0.001 

99.0 

99.0 

11.7 

11.7 

0.002 

98.6 

98.6 

8.6 

98.6 

8.6 

0.005 

98.1 

5.7 

97.5 

5.5 

0.010 

97.2 

96.9 

4.1 

96.3 

4.0 

0.020 

95.7 

3.0 

93.9 

3.0 

0.050 

94.4 

93.8 

1.8 

92.5 

1.9 

0.100 

92.0 

92.0 

1.3 

91.0 

1.4 

*  Jour.  Am.  Chem.  Soc.,  33>  1868  (1911), 


416  THEORETICAL  CHEMISTRY 

Absolute  Velocity  of  the  Ions.  Thus  far  we  have  considered 
only  the  relative  velocities  and  conductances  of  the  ions; 
we  now  proceed  to  the  consideration  of  their  absolute  velocities 
in  centimeters  per  second. 

Let  I  amperes  of  electricity  be  passed  through  a  centimeter 
cube  of  a  solution  of  a  uni-univalent  electrolyte.  If  the  solution 
contains  C  gram-equivalents  of  electrolyte  per  liter,  the  quantity 
of  electricity  carried  by  either  the  cation  or  the  anion  will  be, 

(  Tfi^h)  ^'  wnere  «  is  the  degree  of  ionization,  and  where  F  =  96,500 

coulombs.  Since  the  total  quantity  of  electricity  transported  across 
a  given  section  of  a  solution  of  an  electrolyte  by  its  ions,  is  equal 
to  the  sum  of  the  number  of  equivalents  carried  by  the  cations 
and  the  anions  moving  in  opposite  directions,  it  follows,  that 

U  +  F)  F' 


where  U  and  V  are  the  ionic  velocities  for  unit  potential  gradient. 
But  by  Ohm's  law,  /  =  E/R  =  EL,  hence,  equation  (13)  becomes 


EL  =   lM)   &  +  F>  F> 

or,  for  a  potential  gradient  of  1  volt  per  centimeter, 

L  "  (u  +  F>  F~ 


Substituting  A/A0  for  a,  in  equation  (15),  and  remembering  that 
CA         T 

looo  =  L>  we  have 

Ao  =  (U  +  V)  F.  (16) 

By  means  of  this  equation,  the  velocities  of  the  ions,  expressed  in 
centimeters  per  second,  can  be  calculated,  provided  that  the  corre- 
sponding transference  numbers  are  known.  For  example,  the 
equivalent  conductance  of  a  0.0001  molar  solution  of  potassium 
chloride,  at  18°,  is  128.9;  the  total  velocity  of  the  two  ions  is  then, 

ISO 
9pOO  =  0-001347  cm.  per  sec. 

Phis  total  velocity  is  made  up  of  the  two  individual  ionic  velocities. 
The  transference  numbers  of  the  two  ions,  K*  and  Cl',  are  respec- 


ELECTRICAL  CONDUCTANCE 


417 


tively  0.496  and  0.504.  Hence,  the  absolute  velocities  of  the  ions, 
expressed  in  centimeters  per  second,  in  a  0.001  molar  solution  of 
potassium  chloride,  at  18°,  are  as  follows : 


and 


U  =  0.001347  X  0.496  =  0.00067  cm.  per  sec., 
V  =  0.001347  X  0.504  =  0.00068  cm.  per  sec. 


The  absolute  velocities  of  some  of  the  more  common  ions,  at  18°, 
are  given  in  the  following  table. 

ABSOLUTE  IONIC  VELOCITIES 


Ion. 

Velocity. 

Ion. 

Velocity. 

K* 

cm.  per  sec. 

0  00066 

H* 

cm.  per  sec. 

0  00320 

NH4*  

0.00066 

cr. 

0  00069 

Na*.  . 

O.C0045 

NO3'.  . 

0  00064 

Li* 

0  00036 

CIO/ 

0  00057 

Ag" 

0  00057 

OH' 

0  00181 

CraO7".  .. 

0.000473 

Cu" 

0  00031 

The  velocities  of  certain  ions  have  been  determined  directly. 
Thus,  the  velocity  of  the  hydrogen  ion  was  measured  by  Lodge  * 
in  the  following  manner:  —  The  tube  B,  Fig.  111.  40  cm.  long  and 


Fig.  Ill 

8  cm.  in  diameter,  was  graduated  and  bent  at  right  angles  at  the 
ends.  This  was  filled  with  an  aqueous  solution  of  sodium  chloride 
in  gelatine,  colored  red  by  the  addition  of  an  alkaline  solution  of 
phenolphthalein.  When  the  contents  of  the  tube  had  gelatinized, 
the  tube  was  placed  horizontally,  connecting  two  beakers  filled 
with  dilute  sulphuric  acid,  as  shown  in  the  diagram.  A  current 
of  electricity  was  passed  from  one  electrode  A  to  the  other  elec- 
trode C.  The  hydrogen  ions  from  the  anode  vessel  travelled 
*  Brit.  Assoc.  Report,  p.  393  (1886). 


418  THEORETICAL  CHEMISTRY 

through  the  tube  toward  the  cathode,  discharging  the  red  color  of 
the  phenolphthalein.  By  observing  the  rate  at  which  the  color 
was  discharged,  the  velocity  of  the  hydrogen  ion  under  a  known 
potential  gradient  was  determined.  The  observed  and  calculated 
values  agree  excellently.  It  was  shown  that  the  velocity  of  the 
hydrogen  ion  suffers  almost  no  retardation  from  the  high  viscosity 
of  the  gelatine  solution. 

Whetham,*  in  his  experiments  on  ionic  velocity,  employed 
two  solutions,  one  of  which  possessed  a  colored  ion,  the  motion  of 
the  latter  being  observed  and  its  velocity  determined  under  unit 
potential  gradient.  For  example,  consider  the  boundary  line 
between  two  equally  dense  solutions  of  the  electrolytes,  AC  and 
BC,  C  being  a  colorless,  and  A  a  colored  ion.  When  a  current 
passes  through  the  boundary  between  the  two  electrolytes,  the 
anion,  C,  will  migrate  toward  the  positive  electrode,  while  the  two 
cations,  A  and  B,  will  migrate  toward  the  negative  electrode, 
and  the  color  boundary  will  move  with  the  current,  its  speed  being 
equal  to  that  of  the  colored  ion,  A.  In  this  manner,  Whetham 
measured  the  absolute  velocities  of  the  ions,  Cu",  Cr2(V, 
and  Cl'. 

Ionic  velocities  have  also  been  determined  by  Steele  f  who  ob- 
served the  change  in  the  index  of  refraction  of  the  solution  as  the 
ions  migrated.  The  accompanying  table  gives  a  comparison  of 
the  calculated  and  observed  velocities  of  some  of  the  ions. 

ABSOLUTE  IONIC  VELOCITIES 


Ion, 

Velocity  (obs.). 

Velocity  (calc.). 

HV. 

cm.  per  sec. 
0  002fi 

cm.  per  sec. 

OftAOO 

Cu"  

0  00029 

0  00031 

cr... 

0  00058 

OOOflfiQ 

Cr2O7"  -.    . 

0  00047 

0  0004-73 

Temperature  Coefficient  of  Conductance.  When  the  temper- 
ature of  a  solution  of  an  electrolyte  is  raised,  the  equivalent  con- 
ductance usually  increases.  The  increase  in  conductance  is  due, 
not  to  an  increase  in  the  ionization,  but  to  the  greater  velocity 
of  the  ions.  According  to  Kohlrausch,  the  relation  between  con- 

*  Phil.  Trans.  A.,  184,  337  (1893);  196,  507  (1895). 
t  Phil.  Trans.  A.,  198,  105  (1902). 


ELECTRICAL  CONDUCTANCE 


419 


ductance  and  temperature  may  be  expressed  approximately  by 
the  following  equation, 

A,  =  Ais°  {1  +  j8  (t  -  18)}, 

where  ft  is  the  temperature  coefficient,  or  change  in  conductance 
for  1°  C.     Solving  the  equation  for  ft,  we  have 

At  —  A  is0 


ft 


A18o  (t  -  18) 


(17) 


The  temperature  coefficients  of  several  of  the  more  common  elec- 
trolytes are  given  in  the  accompanying  table. 

TEMPERATURE  COEFFICIENTS  OF  CONDUCTANCE 


Electrolyte. 


Nitric  acid 

Sulphuric  acid 

Hydrochloric  acid 

Potassium  hydroxide 

Potassium  nitrate 

Potassium  iodide 

Potassium  bromide 

Potassium  chlorate 

Silver  nitrate 

Potassium  chloride 

Ammonium  chloride 

Potassium  sulphate 

Copper  sulphate 

Sodium  chloride 

Sodium  sulphate 

Zinc  sulphate 


Temperature 
Coefficient. 


0.0163 
0.0164 
0.0165 
0.0190 
0.0211 
0.0212 
0.0216 
0.0216 
0.0216 
0.0217 
0.0219 
0.0223 
0.0225 
0.0226 
0.0234 
0.0250 


The  temperature  coefficient  of  conductance  is  not,  however,  a 
simple  linear  function  of  the  temperature.  The  following  empiri- 
cal equations,  expressing  equivalent  conductance  at  infinite  dilu- 
tion and  at  any  temperature  t,  in  terms  of  the  conductance  at  18°, 
have  been  derived  by  Kohlrausch: 

Ao,  =  Ao  is"  {1  +  a  (t  -  18)  +  0  (t  -  18)2),  (18) 

and 

ft  =  0.0163  (a  -  0.0174).  (19) 

When  the  values  of  A0 18°,  a,  and  ft,  as  determined  for  a  large  num- 
ber of  electrolytes,  are  substituted  in  the  above  equation,  he 
showed,  that  A0<  becomes  equal  to  zero  at  a  temperature  approx- 


420  THEORETICAL  CHEMISTRY 

imating  to  -40°.  Kohlrausch  suggested  that  each  ion  moving 
through  the  solution  carries  with  it  an  "  atmosphere  "  of  solvent, 
and  that  the  resistance  offered  to  the  motion  of  the  ion  is  simply 
the  frictional  resistance  between  masses  of  pure  water.  This 
view  is  in  harmony  with  the  solvate  theory  discussed  in  an  earlier 
chapter.  Washburn  *  has  calculated  the  degree  of  ionic  hydra- 
tion  for  several  ions.  He  finds,  for  example,  that  the  hydrogen 
ion  carries  with  it  0.3  molecule  of  water,  while  the  lithium  ion  is 
hydrated  to  the  extent  of  4.7  molecules  of  water. 

Conductance  at  High  Temperatures  and  Pressures.  The 
conductance  of  several  typical  electrolytes,  at  temperatures  rang- 
ing from  that  of  the  room  up  to  306°,  have  been  measured  by  A.  A. 
Noyes  and  his  co-workers,  f  These  measurements  were  carried 
out  in  a  conductance  cell  especially  constructed  to  withstand  high 
pressures.  The  results  show  that  the  values  of  A0,  for  binary  elec- 
trolytes, 'tend  to  become  equal  with  rise  of  temperature.  This 
may  be  taken  as  an  indication  of  the  fact  that  the  ionic  velocities 
tend  to  become  equal  as  the  temperature  rises.  The  conduct- 
ance of  ternary  electrolytes  increases  uniformly  with  the  tem- 
perature, and  attains  values  which  are  considerably  greater  than 
those  reached  by  binary  electrolytes.  This  is  what  might  be 
expected,  since  if  an  ion  is  bivalent,  as  in  a  ternary  electrolyte, 
the  driving  force  must  be  greater,  in  consequence  of  which  the 
ion  must  move  faster,  and,  the  conductance  will  increase  cor- 
respondingly. 

The  temperature  coefficient  of  conductance  for  binary  elec- 
trolytes is  greater  between  100°  and  156°,  than  below  or  above 
these  temperatures.  The  temperature  coefficients  of  ternary 
electrolytes  increases  uniformly  with  rising  temperature.  In  the 
case  of  acids  and  bases,  the  rate  of  increase  in  conductance  stead- 
ily diminishes  as  the  temperature  rises.  The  ionization  decreases 
regularly  with  rise  in  temperature,  the  temperature  coefficient  of 
ionization  being  small  between  18°  and  100°. 

The  effect  of  pressure  on  conductance  was  studied  by  Fanjung.f 
He  found  that  the  conductance  increases  slightly  with  increasing 
pressure.  This  result  he  interprets  as  being  due  to  increased 
ionic  velocity,  rather  than  to  an  increase  in  the  number  of  ions 
present  in  the  solution. 

*  Jour.  Am.  Chem.  Soc.,  30,  322  (1909). 

t  Publication  of  Carnegie  Institution,  No.  63. 

i  Zeit.  phys.  Chem.,  14,  673  (1894). 


ELECTRICAL  CONDUCTANCE 


421 


The  Dissociation  of  Water.     Water  behaves  as  a  very  weak 
binary  electrolyte,  dissociating  according  to  the  equation, 

H20  <=»  H'  +  OH'. 

The  specific  conductance  of  water,  purified  with  the  utmost  care, 
has  been  determined  by  Kohlrausch  and  Heydweiller.  *  Their 
results  are  given  in  the  following  table :  — 

SPECIFIC  CONDUCTANCE  OF  WATER 


Temperature, 
degrees. 

Specific  Conduct- 
ance XlO"6. 

0 

0.014 

18 

0.040 

25 

0.055 

34 

0.084 

50 

0.170 

The  conductance  of  pure  water,  at  0°,  is  so  small  that  one  milli- 
meter of  it  has  a  resistance  equal  to  that  of  a  copper  wire  of  the 
same  cross-section  and  40,000,000  kilometers  in  length,  or  in 
other  words,  long  enough  to  encircle  the  earth  one  thousand  times. 
Knowing  the  specific  conductance  of  water,  its  degree  of  dissoci- 
ation can  be  easily  calculated.  The  ionic  conductances  of  the 
two  ions  of  water  at  18°  are  as  follows:  —  H*  =  313,  and  OH'  = 
174.  Therefore,  the  maximum  equivalent  conductance  of  water 
should  be, 

Ao  =  313  +  174  =  487. 

The  equivalent  conductance,  at  18°,  of  a  liter  of  water  between 
electrodes  1  cm.  apart  is,  according  to  the  data  of  Kohlrausch, 

0.04  X  10-6  X  103  =  0.04  X  10-3; 
therefore, 

0  04  X  10~3 

--  =  0.8  X  10~7  =  c,  the  concentration  of  the  ions, 


H'  and  OH',  in  mols  per  liter  at  18°. 

Conductance  of  Difficultly-Soluble  Salts.  In  a  saturated 
solution  of  a  difficultly-soluble  salt,  the  solution  is  so  dilute  that,  in 
general,  we  may  assume  complete  ionization,  or  A  =  A0. 

*  Zeit.  phys.  Chem.,  14,  317  (1894). 


422  THEORETICAL  CHEMISTRY 

When  this  is  the  case,  we  have 

i/solution   ~  LHZO   =  L, 

and 

1000  L 
A  =  A0  =    —  £  — 

Hence, 

„      1000  L 

V   =    "~7  -  ' 

AO 

where  C  denotes  the  concentration  in  gram-equivalents  per  liter. 
Thus,  Bottger  found  for  a  saturated  solution  of  silver  chloride,  at 
20°,  L'  =  1.374  X  10~6.  Deducting  the  specific  conductance  of 
the  water  at  this  temperature,  we  have 

L  =  1.374  X  10-6  -  0.044  X  10~6  =  1.33  X  lO"6. 
Since  the  value  of  A0,  at  20°,  for  silver  chloride,  determined  from 
the  table  of  ionic  conductances,  is  119.2,  we  have 


m  =  100°X11i'noX  =  1-12  X  10-5  gr.-equiv.  AgCl  per  liter. 

ny.z 

The  Conductance  of  Pure  Liquids.  In  general,  the  conductance 
of  pure  liquids  is  small.  Thus,  the  specific  conductance  of  pure 
water,  at  18°,  is  approximately  1  X  10~6  reciprocal  ohms  and,  as 
Walden  *  has  shown,  the  specific  conductance  of  a  number  of 
other  solvents  is  of  the  same  order  as  that  for  water.  Mixtures 
of  two  liquids,  each  of  which  is  practically  non-conducting,  may 
have  a  conductance  differing  but  little  from  that  of  the  two  com- 
ponents; or,  the  mixture  may  have  a  very  high  conductance.  For 
example,  the  conductance  of  a  mixture  of  water  and  ethyl  alcohol 
is  of  the  same  order  of  magnitude  as  that  of  the  two  components, 
while  on  the  other  hand,  a  mixture  of  water  and  sulphuric  acid, 
each  of  which,  in  the  pure  state,  is  practically  a  non-conductor, 
has  great  conducting  power.  The  variation  of  the  specific  con- 
ductance of  mixtures  of  water  and  sulphuric  acid  is  represented 
graphically  in  Fig.  112,  concentration  being  plotted  on  the 
axis  of  abscissae  and  specific  conductance  on  the  axis  of  or- 
dinates.  It  appears,  that  as  the  concentration  of  the  sulphuric 
acid  increases,  the  specific  conductance  of  the  mixture  increases 
until  30  per  cent  of  acid  is  present,  beyond  which  point  it  grad- 
ually diminishes.  When  pure  sulphuric  acid  is  present,  the  value 

*  Zeit.  phys.  Chem.,  46,  103  (1903). 


ELECTRICAL  CONDUCTANCE  423 

of  the  specific  conductance  is  practically  zero.  On  dissolving 
sulphur  trioxide  in  the  pure  acid,  the  specific  conductance  in- 
creases slightly  to  a  maximum,  and  then  falls  rapidly  to  zero. 
There  is  a  minimum  in  the  curve  corresponding  to  about  85  per 
cent  of  acid,  a  concentration  which  corresponds  almost  exactly 


20         30         40         60         60         70         80         90        100        110 
Per  Cent  Sulphuric  Add 

Fig.  112 

with  the  hydrate,  H2S04'H20.  Why  some  liquid  mixtures  should 
have  marked  conducting  power  and  others  hardly  any,  it  is  diffi- 
cult to  explain.  Many  fused  salts,  such  as  silver  nitrate  and  lith- 
ium chloride,  are  excellent  conductors  and  are  thus  exceptions 
to  the  general  rule,  that  pure  substances,  belonging  to  the  second 
class  of  conductors,  possess  little  conducting  power. 

Conductance  of  Fused  Salts.  While  solid  salts  are  exceed- 
ingly poor  conductors  of  electricity,  yet  as  the  temperature  is 
raised  their  conductance  increases,  until,  at  their  melting-points, 
they  become  good  conductors.  The  specific  conductance  of  a 
fused  salt  may  even  exceed  the  specific  conductance  of  the 
most  concentrated  aqueous  solutions,  but  owing  to  its  high 
concentration,  the  equivalent  conductance  is  much  less.  The 
following  table  gives  the  specific  and  equivalent  conductance  of 
fused  silver  nitrate. 


424 


THEORETICAL  CHEMISTRY 


Temperature, 
degrees. 

Sp.  Cond. 

Equiv.  Cond. 

218  (melt.-pt.) 

0.681 

29.2 

250 

0.834 

36.1 

300 

1.049 

46.2 

350 

1.245 

55.4 

The  specific  conductance  of  a  60  per  cent  aqueous  solution  of 
silver  nitrate,  at  18°,  is  0.208  reciprocal  ohm. 

If  the  salts  are  impure  the  conductance  is  raised,  the  effect  of 
impurities  being  apparent  even  before  the  salts  have  reached  their 
melting-points.  This  is  analogous  to  the  behavior  of  solutions, 
and  suggests  that  the  impurity  functions  in  the  salt  mixture  as  a 
dissolved  solute.* 

Conductance  of  Non-aqueous  Solutions.  A  large  amount  of 
interesting  and  important  work  has  been  done  in  recent  years 
on  the  electrical  conductance  of  solutions  in  non-aqueous  sol- 
vents. In  general,  those  liquids  which  yield  conducting  solutions 
are  themselves  also  conductors  of  electricity.  When  the  solvent 
is  an  inorganic  substance,  such  as  ammonia,  sulphur  dioxide  or 
hydrocyanic  acid,  the  conductance  is  comparable  with  that  of 
aqueous  solutions,  but  when  the  solvent  is  an  organic  compound, 
the  conductance  is  almost  always  very  much  less  than  that  of  the 
corresponding  aqueous  solution.  The  following  examples  will 
serve  to  illustrate  the  wide  variation  in  the  conductance  of  solu- 
tions of  silver  nitrate  in  different  solvents. 

CONDUCTANCE  OF  SILVER  NITRATE  IN 
DIFFERENT  SOLVENTS! 


Temp.  25  < 


SILVER  NITRATE  IN  ETHYL 
ALCOHOL 

C  A26° 

0.1  10.81 

0.01  22.06 

0.0078  23.72 

0.003  28.14 

0.001  34.95 

0.00056  37.79 

0.0003  38.52 

0.0001  40.71 


SILVER  NITRATE  IN  METHYL 

ALCOHOL 

c  A26° 

0.1  36.64 

0.01  68.75 

0.006  74.88 

0.003  83.14 

0.0012  87.30 

0.001  90.10 

0.0003  87.82 

0.0001  73.23 


*  For  a  complete  treatment  of  fused  electrolytes,  the  student  is  advised 
to  consult,  "  Die  Elektrolyse  geschmolzener  Salze,"  by  Richard  Lorenz. 
t  Gibbons  &  Getman.  Jour.  Am.  Chem.  Soc.  36,  1640  (1914). 


ELECTRICAL  CONDUCTANCE  425 

SILVER  NITRATE  IN  ACETONE  SILVER  NITRATE  IN  ANILINE 


C  A25° 

0.01  10.51 

0.003  11.39 

0.001  15.43 

0.0003  20.78 

0.00016  25.60 

0.0001  28.06 


c  A25° 

0.1  0.666 

0.01  0.327 

0.003  0.436 

0.001  0.678 

0.0003  1.082 

0.0001  1  651 


SILVER  NITRATE  IN  PYRIDINE 


c  A26C 

0.1  24.80 

0.01  33.85 

0.003  43.68 

0.001  54.13 


c  A26° 

0.0003  66  14 

0.0001  73.50 

0.00006  75.82 


The  conductance  of  alcoholic  solutions  is  found  to  dimmish  as  the 
molecular  weight  of  the  solvent  increases.  In  fact,  in  any  homol- 
ogous series  whose  members  form  conducting  solutions,  the  con- 
ductance decreases  as  the  number  of  carbon  atoms  in  the  solvent 
increases. 

A  study  of  solutions  in  anhydrous  formic  acid  has  recently 
been  made  by  Schlesinger  *  who  finds  that  these  solutions  are  more 
nearly  normal  in  their  behavior  than  the  corresponding  aqueous 
solutions.  Solutions  of  potassium  iodide  in  acetone  conduct 
better  than  the  corresponding  solutions  in  methyl  and  ethyl  alco- 
hols, whereas,  the  conductance  of  lithium  chloride  and  silver 
nitrate  in  the  latter  solvents  is  much  greater  than  in  acetone. 
Solutions  in  pyridine  are  found  to  conduct  well,  while  solutions 
in  the  analogous  solvents,  piperidine  and  quinoline,  are  to  be 
classed  among  poor  conductors.  Such  irregularities  in  the 
conductance  of  the  same  electrolyte  in  different  non-aqueous 
solvents  are  found  to  be  quite  general. 

In  nearly  every  instance,  the  equivalent  conductance  increases 
with  the  dilution,  as  in  the  case  of  aqueous  solutions.  There  are 
numerous  exceptions  to  this  rule,  however,  a  notable  example 
being  that  of  silver  nitrate  in  aniline,  the  conductance  curve  of 
which  is  shown  in  Fig.  113. 

The  molecular  weight  of  the  solute  in  dilute  aqueous  solutions, 
as  determined  by  the  freezing-point  and  boiling-point  methods, 

*  Jour.  Am.  Chem.  Soc.,  41,  1923  (1919). 


426 


THEORETICAL  CHEMISTRY 


is  usually  much  less  than  the  simple  formula-weight.  In  non- 
aqueous  solutions,  the  reverse  is  quite  frequently  found  to  be 
true. 

Owing  to  the  difficulty  in  determining  A0  with  sufficient  accu- 
racy, it  is  impossible,  in  most  cases,  to  calculate  the  degree  of 
ionization  of  an  electrolyte  in  non-aqueous  solution.  In  those 


o  Gibbons  and  Getman 
•  Sachanov 


10 


15 


20 


-y  Volume 

Fig.1 113 

few  cases  where  it  has  been  possible  to  estimate  the  value  of  AQ, 
the  degree  of  ionization,  thus  calculated,  has  rarely  been  found 
to  agree  with  that  calculated  from  freezing-point  or  boiling-point 
measurements. 

While  solutions  of  electrolytes  in  the  lower  alcohols  resemble 
aqueous  solutions  very  closely,  this  resemblance  is  found  to  de- 
crease rapidly  as  the  molecular  weight  of  the  alcohol  increases. 
Solutions  of  electrolytes  in  almost  all  other  non-aqueous  solvents 
do  not  appear  to  undergo  dissociation  in  the  same  manner  as  in 
aqueous  solutions.  In  fact,  it  is  known  that  the  molecules  of 
electrolytes  tend  to  become  polymerized  in  non-aqueous  solutions. 
These  polymerized  molecules  may  be  formed  by  the  union  of  two 
or  more  molecules  of  the  solute,  or  by  the  combination  of  one  or 
more  molecules  of  the  solute  with  one  or  more  molecules  of  the 
solvent.  Undoubtedly,  it  is  to  the  presence  of  such  aggregates 
in  certain  non-aqueous  solutions  that  the  decrease  in  their 
conductance  on  dilution  is  to  be  ascribed. 

Ionizing  Power  of   Solvents.     Both  Thomson*  and  Nernst  f 
*  Phil.  Mag.,  36,  320  (1893). 
t  Zeit.  phys.  Chem.,  13,  531  (1894). 


ELECTRICAL  CONDUCTANCE 


427 


pointed  out,  that  if  the  forces  which  hold  the  atoms  in  the  mole- 
cule are  of  electrical  origin,  then  those  liquids  which  possess  large 
dielectric  constants  should  have  correspondingly  great  ionizing 
power.  This  is  a  direct  consequence  of  Coulomb's  law  of  electro- 
static attraction,  which  may  be  expressed  by  the  equation, 


(20) 


in  which  q±  and  q%  denote  the  electric  charges,  d  the  distance 
between  them,  /  the  force  of  attraction  and  K  the  dielectric  con- 
stant. Obviously  the  larger  K  becomes,  the  smaller  will  be  the 
value  of  /;  i.e.,  the  more  likely  the  molecule  will  be  to  break  down 
into  ions.  That  the  above  relation  is  approximately  true  may  be 
seen  from  the  following  table. 


DIELECTRIC  CONSTANTS 


Solvent. 

K 

Ionizing  Power. 

Benzene                                  

2  3 

Extremely  weak 

Ethyl  ether  

4.1 

Weak 

Ethyl  alcohol  

25 

Fairly  strong 

Formic  acid  

62 

Strong 

Water 

80 

Very  strong 

Hydrocyanic  acid 

96 

Very  strong 

Dutoit  and  Aston  *  have  suggested  that  there  is  a  connection 
between  the  ionizing  power  of  a  solvent  and  its  degree  of  associa- 
tion, while  Dutoit  and  Friderich  f  conclude  from  their  experiments 
that  the  values  of  A0  for  a  given  electrolyte  dissolved  in  different 
solvents,  are  a  direct  function  of  the  degree  of  association,  and 
an  inverse  function  of  the  viscosity  of  the  solvents.  Water  and 
the  alcohols  furnish  good  illustrations  of  the  truth  of  this  general- 
ization. 

REFERENCES 

Text-Book  of  Electrochemistry.     Le  Blanc.     Translated  by  Whitney  and 

Brown.     Chapters  I  to  VI  incl. 
Electrochemistry.     Lehfeldt.     Chapters  I  &  II. 
Conductivity  of  Liquids.     Tower. 

*  Compt.  rend.,  125,  240  (1897). 

f  Bull.  Soc.  Chim.  [3],  19,  321  (1898). 


428  THEORETICAL  CHEMISTRY 

Applied  Electrochemistry.     Allmand.     Chapters  I  to  VI  incl. 
Applied  Electrochemistry.     Thompson.     Chapter  I. 

Electrical  Conductivity  and  lonization  Constants  of  Organic  Compounds. 
Scudder. 

PROBLEMS 

1.  An  aqueous  solution  of  copper  sulphate  is  electrolyzed  between 
copper  electrodes  until  0.2294  gram  of  copper  is  deposited.     Before  elec- 
trolysis the  solution  at  the  anode  contained   1.1950  grams  of  copper, 
after  electrolysis  1.3600  grams.     Calculate  the  transference  numbers  of 
the  two  ions,  Cu"  and  SO/7.  Am.  n  =  0.72     1  -  n=  0.28. 

2.  A  solution  containing  0.1605  per  cent  of  NaOH  was  electrolyzed 
between   platinum   electrodes.     After   electrolysis   55.25   grams   of   the 
cathode  solution  contained  0.09473  gram  of  NaOH,  whilst  the  concen- 
tration of  the  middle  portion  of  the  electrolyte  was  unchanged.     In  a 
silver  coulometer  the  equivalent  of  0.0290  gram  of  NaOH  was  deposited 
during  electrolysis.     Calculate  the  transference  numbers  of  the  Na*  and 
OH'  ions. 

3.  In  a  0.01  molar  solution  of  potassium  nitrate,  the  transference 
numbers  of  the  cation  and  anion  are,  respectively,  0.503  and  0.497.     Find 
the  equivalent  conductances  of  the  two  ions  in  this  solution  having  given 
that  its  specific  conductance  is  0.001044.     Ans.  lc  =  52.5,      la  =  51.9. 

4.  The  absolute  velocity  of  the  Ag*  ion  is  0.00057  cm.  per  sec.,  and  that 
of  the  Cl'  ion  is  0.00069  cm.  per  sec.     Calculate  the  equivalent  conduct- 
ance of  an  infinitely  dilute  solution  of  silver  chloride. 

5.  The  equivalent  conductance  of  an  infinitely  dilute  solution  of  am- 
monium chloride  is  130;   the  ionic  conductances  of  the  ions  OH'  and  Cl' 
are  174  and  65.44  respectively.     Calculate  the  equivalent  conductance 
of  ammonium  hydroxide  at  infinite  dilution.  Ans.  Ao  =  238.56. 

6.  The  equivalent  conductance  of  a  molar  solution  of  sodium  nitrate 
at  18°  is  66;  its  conductance  at  infinite  dilution  is  105.3.    What  is  the 
degree  of  ionization  in  the  molar  solution? 

Q)  The  specific  conductance  of  a  saturated  solution  of  AgCN  at  20° 
is  1.79  X  10~6  and  the  specific  conductance  of  water  at  the  same  temper- 
ature is  0.044  X  10-6  reciprocal  ohms.  The  equivalent  conductance  at 
infinite  dilution  is  115.5.  Calculate  the  solubility  of  AgCN  in  grams  per 
liter. 

8.  The  equivalent  conductance  at  18°  of  a  solution  of  sodium  sulphate 
containing  0.1  gram-equivalent  of  salt  per  liter  is  78.4,  the  conductance 
at  infinite  dilution  is  113  reciprocal  ohms.  What  is  the  value  of  i  for 
the  solution?  What  is  its  osmotic  pressure? 


ELECTRICAL  CONDUCTANCE  429 

9.  The  freezing-point  of  a  0.1  molar  solution  of  CaCl2  is  -0°.482. 
'  (a)  Calculate  the  degree  of  ionization  (freezing-point  constant  =  1.89 
for  one  mol  per  liter),  (b)  Calculate  the  degree  of  ionization  from  the 
equivalent  conductance  at  18°,  which  is  82.79  reciprocal  ohms,  whilst 
the  equivalent  conductance  of  CaCl2  at  infinite  dilution  is  115.8  reciprocal 
ohms.  Ans.  (a)  «  =  0.774,  (b)  «  =  0.715 


CHAPTER  XVI 
ELECTROLYTIC  EQUILIBRIUM   AND   HYDROLYSIS 

Ostwald's  Dilution  Law.  It  has  been  shown,  in  preceding 
chapters,  that  the  law  of  mass  action  is  applicable  to  chemical 
equilibria,  in  both  gaseous  and  liquid  systems.  We  now  proceed 
to  show  that  it  applies  equally  to  electrolytic  equilibria.  When 
acetic  acid  is  dissolved  in  water  it  dissociates  according  to  the 
equation, 

CHsCOOH  <=>  CH3CO(y  +  H\ 

Let  one  mol  of  acetic  acid  be  dissolved  in  water,  and  the  solution 
diluted  to  v  liters,  and  let  a  denote  the  degree  of  dissociation. 

Then,  the  concentration  of  the  undissociated  acid  is  -      — »  and 

9 

the  concentration  of  each  of  the  ions  is  -  •     Applying   the  law 
of  mass  action,  we  have 


(!-«)»  "  K'  (1} 

where  K  is  the  equilibrium,  or  ionization  constant. 

This  equation  expressing  the  relation  between  the  degree  of 
ionization  and  dilution,  was  derived  by  Ostwald,*  and  is  known 

as  the  Ostwald  dilution  law.     Since  a  =  — ,  we  may  substitute 

AO 

this  value  of  a  in  equation  (1),  and  obtain  the  expression, 

A2 


—  A)  v 


K.    -  (2) 


The  dilution  law  may  be  tested  by  substituting  the  value  of  a, 
corresponding  to  any  dilution  v,  in  equation  (1)  and  calculating 

*  Zeit.  phys.  Chem.,  2,  36  (1888);  3,  170  (1889). 
430 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS        431 

the  value  of  the  ionization  constant,  K;  the  value  of  a,  at  any 
other  dilution,  may  then  be  calculated,  and  compared  with  the 
value  determined  by  direct  experiment.  The  following  table  gives 
the  results  obtained  with  acetic  acid  at  14°.  1,  K  being  equal  to 
0.0000178. 

DEGREE  OF  IONIZATION  OF  ACETIC  ACID 


v  (in  liters). 

aXlQz  (calc.). 

«X1Q2  (obs.). 

0.994 

0.42 

0.40 

2.02 

0.60 

0.614 

15.9 

1.67 

1.66 

18.1 

1.78 

1.78 

1,500.0 

15.0 

14.7 

3,010.0 

20.2 

20.5 

7,480.0 

30.5 

30.1 

15,000.0 

40.1 

40.8 

As  will  be  seen,  the  agreement  between  the  observed  and  cal- 
culated values  is  very  close.  The  table  also  shows,  to  how  small 
an  extent  the  molecules  of  acetic  acid  are  broken  down  into  ions, 
a  molar  solution  being  dissociated  less  than  0.5  per  cent.  The 
dilution  law  holds  for  nearly  all  organic  acids  and  bases,  but  fails 
to  apply  to  salts,  strong  acids,  and  strong  bases.  When  a  is 
small,  the  term  (1  —  a)  does  not  differ  appreciably  from  unity, 
and  equation  (1)  becomes, 


or 


a  =  VvK. 


(3) 


On  the  other  hand,  when  a  cannot  be  neglected,  we  have,  on 
solving  equation  (1)  for  a, 


=  --^-  +  \  vK 


(4) 


The  method  of  derivation  indicates  that  the  dilution  law  is  only 
strictly  applicable  to  binary  electrolytes,  and  therefore,  it  is  im- 
probable that  it  will  hold  for  electrolytes  yielding  more  than  two 
ions.  It  has  been  found,  however,  that  organic  acids,  whether 
they  are  mono-,  di-,  or  polybasic,  always  ionize  as  a  monobasic 
acid  up  to  the  dilution  at  which  «  =  50  per  cent.  This  means, 


432  THEORETICAL  CHEMISTRY 

that  the  dilution  law  is  applicable  to  polybasic  acids  up  to  that 
dilution  at  which  the  acid  is  50  per  cent  ionized. 

Strength  of  Acids  and  Bases.  There  are  several  methods  by 
which  the  relative  strengths  of  acids  can  be  estimated.  A  method 
which  has  proved  of  great  value  is  that  in  which  two  different 
acids  are  allowed  to  compete  for  a  certain  base,  the  amount  of 
which  is  insufficient  to  saturate  both  of  them.  Suppose  equiva- 
lent weights  of  nitric  and  dichloracetic  acids  to  be  mixed  together 
with  sufficient  potassium  hydroxide  to  saturate  one  acid  com- 
pletely. In  order  to  determine  the  position  of  the  equilibrium 
represented  by  the  equation, 

HN3  .  +  CHC12  •  COOK  <=±  CHC12  •  COOH  +  KNO3, 

we  may  make  use  of  any  method  which  does  not  disturb  the  equi- 
librium. Since  ordinary  chemical  methods  are  excluded  on  this 
account,  it  is  customary  to  determine  the  change  in  some  physical 
property  accompanying  the  reaction.  Thus,  Ostwald*  found  that 
when  one  mol  of  potassium  hydroxide  is  neutralized  by  nitric  acid 
in  dilute  solution,  the  volume  increases  approximately  20  cc.  When 
one  mol  of  potassium  hydroxide  is  neutralized  by  dichloracetic  acid, 
however,  the  increase  in  volume  is  13  cc.  It  is  evident,  therefore, 
that  if  nitric  acid  completely  displaces  dichloracetic  acid,  as  rep- 
resented by  the  above  equation,  the  increase  in  volume  will  be, 
20— 13  =  7  cc. ;  if  no  displacement  occurs,  however,  then  the  volume 
will  remain  constant.  He  found,  that  the  volume  actually  increased 
5.67  cc.  Therefore,  the  reaction  represented  by  the  upper  arrow, 
has  proceeded  to  the  extent  of  5.67  -f-  7  =  80  per  cent.  That  is 
to  say,  in  the  competition  of  the  two  acids  for  the  base,  the  nitric 
acid  has  taken  80  per  cent,  and  the  dichloracetic  acid  has  taken 
20  per  cent,  or  the  relative  strengths  of  the  two  acids  are  in  the 
ratio  of  80  :  20,  or  4  :  1. 

The  relative  strengths  of  acids  can  also  be  determined  from  their 
catalytic  effect  on  the  rates  of  certain  reactions,  such  as  the  hy- 
drolysis of  esters  or  the  inversion  of  cane  sugar. 

The  order  of  the  activity  of  acids  is  the  same  whether  measured 
by  equilibrium  or  kinetic  methods.  Arrhenius  pointed  out,  that 
the  relative  strengths  of  acids  can  be  readily  determined  from 
their  electrical  conductance.  The  order  of  the  strengths  of  acids, 
as  determined  by  equilibrium  and  kinetic  methods,  is  the  same  as 
*  Jour,  prakt.  Chem.  [2],  i8,,328  (1878). 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS        433 


that  of  their  electrical  conductances  in  equivalent  solutions. 
This  is  well  illustrated  by  the  following  table  in  which  the  three 
methods  are  compared,  hydrochloric  acid  being  taken  as  the 
standard  of  comparison. 

RELATIVE  STRENGTH  OF  ACIDS 


Acid. 

Method  Employed. 

Equilibrium. 

Kinetic. 

Conductance. 

HC1 

100 
100 
49 
9 

100 

100 
53.6 
4.8 
0.4 

100 
99.6 
65.1 
4.8 
1.4 

HNO3  

H2SO4  .... 

CH2C1COOH   . 

CHgCOOH  

The  results  of  these,  and  other  experiments,  warrant  the  con- 
clusion that  the  strength  of  an  acid  is  determined  by  the  number 
of  hydrogen  ions  which  it  yields.  It  is  important  to  note,  that  the 
electrical  conductance  ol  an'acid  is  not  directly  proportional  to 
its  hydrogen  ion  concentration:  the  relatively  high  velocity  of 
the  H'  ion  is  the  cause  of  the  approximate  proportionality  between 
these  two  variables.  In  the  case  of  a  weak  acid,  the  value  of  the 
ionization  constant  may  be  taken  as  a  measure  of  its  strength. 
The  following  table  gives  the  values  of  the  ionization  constants, 
at  25°,  for  several  different  acids. 

IONIZATION  CONSTANTS  OF  ACIDS 


Acid. 

Ionization 
Constant. 

Acetic  acid  

0.0000180 

Monochloracetic  acid  
Trichloracetic  acid  . 

0.00155 
1  21 

Cyanacetic  acid  

0.0037 

Formic  acid  

0.000214 

Carbonic  acid  

3040  XHT10 

Hydrocyanic  acid 

570  XlO"10' 

Hydrogen  sulphide  
Phenol                               

13X10-10! 
1.3X10~10 

Since  for  a  weak  acid,  a  =  Vv  K,  it  follows,  that  for  two  weak 
acids  at  the  same  dilution,  we  may  write, 


01 


(5) 


434 


THEORETICAL  CHEMISTRY 


or,  the  ratio  of  the  degrees  of  ionization  of  the  two  acids  is  equal  to 
the  square  root  of  the  ratio  of  their  ionization  constants.  Thus, 
from  the  data  given  in  the  foregoing  table  for  acetic  and  mono- 
chloracetic  acids,  we  have 


ai       t/ 

5  =  V 


1 


0.000018 


0.00155       9.3 


or,  the  effect  of  replacing  one  atom  of  hydrogen  in  the  methyl 
group  of  acetic  acid  increases  the  strength  of  the  acid  about  nine 
times. 

Just  as  the  hydrogen  ion  concentration  of  acids  determines  their 
strength,  so  the  strength  of  bases  is  determined  by  the  concen- 
tration of  hydroxyl  ions.  The  strength  of  bases  may  be  estimated 
by  methods  similar  to  those  employed  in  determining  the  strength 
of  acids.  Thus,  two  different  bases  may  be  allowed  to  compete 
for  an  amount  of  acid  sufficient  to  saturate  only  one  of  them;  or 
a  catalytic  method,  developed  by  Koelichen,*  may  be  used.  This 
method  is  based  upon  the  effect  of  hydroxyl  ions  on  the  rate  of 
3ondensation  of  acetone  to  diacetonyl  alcohol,  as  represented  by 
the  equation, 

2  CH3COCH3  =  CH3COCH2C  (CH3)2OH. 

In  addition  to  these  two  methods,  the  method  of  electrical  con- 
ductance is  also  applicable.  The  agreement  between  the  results 
obtained  by  the  three  methods  is  quite  satisfactory.  The  alkali 
and  alkaline  earth  hydroxides  are  very  strong  bases,  and  are  dis- 
sociated to  about  the  same  extent  as  equivalent  solutions  of 
hydrochloric  and  nitric  acids,  while  on  the  other  hand,  ammonia 
and  many  of  the  organic  bases  are  very  weak.  The  following 
table  gives  the  ionization  constants  of  several  typical  bases. 

IONIZATION  CONSTANTS  OF  BASES 


Base. 

Ionization 
Constant. 

Ammonia  r.  . 

0  000023 

Me  t  hy  1  am  i  ne 

0  00050 

Trimethylamine 

0  000074 

Pyridine  

2  5X10~10 

Aniline  

1  1X10~10 

*  Zeit.  phys.  Chem.,  33,  129  (1900). 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS        435 

Mixtures  of  Two  Electrolytes  with  a  Common  Ion.  Just  as 
the  dissociation  of  a  gaseous  substance  is  diminished  by  the  addi- 
tion of  an  excess  of  one  of  the  products  of  dissociation,  so  the 
ionization  of  weak  acids  and  bases  is  depressed  by  the  addition  of  a 
salt,  having  an  ion  common  to  the  acid,  or  the  base.  If  the  degree 
of  ionization  of  a  salt  having  an  ion  in  common  with  an  acid,  or  a 
base,  is  represented  by  a ',  and  if  n  denotes  the  number  of  molecules 
of  salt  present,  then  the  equation  of  equilibrium  of  the  acid,  or 
base  will  be 

(not'  +  a)  a  =  Kv  (1  -  a),  (6) 

where  a  is  the  degree  of  ionization  of  the  acid,  or  base.  For  very 
weak  acids  and  bases,  a  is  so  small,  that  1  —  a  does  not  differ 
appreciably  from  unity,  and  since  a!  is -practically  independent  of 
the  dilution,  we  obtain, 

na  =  Kv  (7) 

or  «=**.  (8) 

n 

That  is,  the  ionization  of  a  weak  acid,  or  base,  in  the  presence  of 
one  of  its  salts,  is  approximately  inversely  proportional  to  the 
amount  of  salt  present. 

In  many  of  the  processes  of  analytical  chemistry,  advantage  is' 
taken  of  the  action  of  neutral  salts  on  the  ionization  of  weak  acids 
and  bases.  Thus,  while  the  concentration  of  hydroxyl  ions  in 
ammonium  hydroxide  is  sufficient  to  precipitate  magnesium  hy- 
droxide from  solutions  of  magnesium  salts,  the  presence  of  a  small 
amount  of  ammonium  chloride  depresses  the  ionization  of  the 
ammonium  hydroxide  to  such  an  extent  that  precipitation  no 
longer  takes  place. 

Isohydric  Solutions.  Arrhenius*  was  the  first  to  point  out  what 
relation  must  exist  between  solutions  of  two  electrolytes  having 
a  common  ion,  in  order  that,  when  mixed  in  any  proportions, 
they  may  not  exert  any  mutual  influence.  He  showed  that  when 
the  concentration  of  the  common  ion,  in  each  of  the  two  solutions, 
is  the  same  before  mixing,-  no  alteration  in  the  degree  of  ionization 
will  occur  after  mixing.  Such  solutions  are  said  to  be  isohydric. 
Thus,  an  aqueous  solution  containing  one  mol  of  acetic  acid  in 
8  liters,  is  isohydric  with  an  aqueous  solution  containing  one  mol 
of  hydrochloric  acid  in  667  liters.  On  mixing  these  two  solutions, 
the  hydrogen  ion  concentration  remains  unchanged,  and  if  the 
*  Wied.  Ann.,  30,  51  (1887). 


436  THEORETICAL  CHEMISTRY 

mixture  is  treated  with  a  small  amount  of  sodium  hydroxide, 
equal  amounts  of  sodium  acetate  and  sodium  chloride  will  be 
formed. 

That  isohydric  solutions  may  be  mixed  without  altering  their 
respective  ionizations,  may  be  shown  in  the  following  manner  :  — 
Let  C  and  c  denote  the  concentrations  of  the  undissociated  por- 
tions, and  let  CA,  Cz,  CA,  and  c^  denote  the  concentrations  of  the 
dissociated  portions  of  two  electrolytes,  while  €2  and  c^,  correspond 
to  two  different  ions. 

Then,  we  have 

kc  =  cAC2,  (9) 

and  KC  =  CAC2.  (10) 

If  V  liters  of  the  first  solution  be  mixed  with  v  liters  of  the  second 
solution,  the  concentrations  of  the  undissociated  portions,  and  of 
the  dissimilar  ions,  will  be 

CV  cv  C2V  w 

TTV        v  +  v'        FTV  v  +  v' 

while  the  concentration  of  the  common  ion  A,  becomes 


Applying  the  law  of  mass  action,  we  have 

*<  =  c4r^. 

and 

Kr      CAV  +  cAvr  .    . 

KC  =        V  +  v    '    *' 

But  equations  (11)  and  (12)  only  become  identical  with  equations 
(9)  and  (10)  when  CA  =  CA,  or,  in  other  words,  no  change  in  the 
degree  of  dissociation  takes  place  after  the  two  solutions  are  mixed. 
lonization  of  Strong  Electrolytes.  It  has  already  been  men- 
tioned that  the  Ostwald  dilution  law,  which  is  a  direct  conse- 
quence of  the  law  of  mass  action,  applies  only  to  weak  electrolytes. 
Just  why  the  law  of  mass  action  should  fail  to  apply  to  strong 
electrolytes  is  not  known,  but  several  possible  causes  have  been 
suggested  to  account  for  its  failure.  One  of  the  most  plausible 
explanations  is  that  advanced  by  Biltz,*  who  attributes  its  failure 
to  hydration  of  the  solute.  If  the  ions  become  associated  with 

*  Zeit.  phys.  Chem.,  40,  218  (1902). 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS         437 


a  large  proportion  of  the  solvent,  the  effective  ionic  concentration 
would  then  be,  the  ratio  of  the  amount  of  the  ion  present  to  the 
amount  of  the  free  solvent,  instead  of  to  the  total  amount  of  sol- 
vent, as  ordinarily  calculated.  This  view  is  in  harmony  with  cer- 
tain facts  which  have  been  adduced  in  favor  of  the  theory  of 
hy  drat  ion. 

While  the  Ostwald  dilution  law  does  not  apply  to  strongly  ion- 
ized electrolytes,  certain  empirical  expressions  have  been  derived 
which  hold  fairly  well  over  a  wide  range  of  dilution.  Thus, 
Rudolphi  showed  that  the  equation, 


(1  -  a) 


(13) 


gives  approximately  constant  values  for  K',  with  strong  electro- 
lytes. The  following  table  gives  the  results  obtained  with  solu- 
tions of  silver  nitrate,  at  25°;  the  numbers  in  the  third  column 
were  calculated  by  means  of  the  Ostwald  dilution  law,  while 
those  in  the  fourth  column  were  calculated  by  means  of  Rudolphi's 
dilution  law. 

COMPARISON  OF  OSTWALD   &   RUDOLPHI  EQUATIONS 


V 

a 

K 

K' 

r 

16 

0.8283 

0.253 

1.11 

32 

0.8748 

0.191 

1.16 

64 

0.8993 

0.127 

1.06 

128 

0.9262 

0.122 

1.07 

256 

0.9467 

0.124 

1.08 

512 

0.9619 

0.125 

1.09 

The    Rudolphi    equation   was    modified    by    van't    Hoff    as 
follows, 


=  K' 


(14) 


(1  -  a 

This  equation  holds  even  more  closely  than  that  of  Rudolphi. 
Of  the  more  recent  empirical  equations  which  have  been  derived 
to  express  the  change  of  conductance  of  an  electrolyte  with  dilu- 
tion, the  equations  of  Kraus  *  and  Bates  f  deserve  special  men- 
tion. 

*  Jour.  Am.  Chem.  Soc.,  35,  1412  (1913). 

t  Ibid.,  37,  1421  (1915). 


438 


THEORETICAL  CHEMISTRY 


The  equation  of  Kraus  has  the  following  form: 


\AoW 


L  _     AT? 
\          Aorjo, 


(15) 


In  this  equation,  A  is  the  conductance  of  the  electrolyte  whose  con- 
centration, C,  is  expressed  in  mols  per  liter,  77/770  is  the  ratio  of  the 
viscosity  of  the  solution  to  that  of  the  solvent,  and  k,  k',  h,  and  A0 
are  empirical  constants,  the  values  of  which  are  so  chosen  as  to 
insure  close  agreement  between  the  observed  and  calculated  values 
of  the  conductance. 

The  equation  of  Bates  is  similar  to  that  of  the  preceding  equa- 
tion, except  that  the  logarithm  of  the  left-hand  side  of  the  equation 
is  substituted  for  the  original  expression  of  Kraus.  The  equation 
of  Bates  takes  the  form: 


.  (16) 


The  constants,  k,  k',  and  h,  are  purely  empirical,  as  in  the  equation 
of  Kraus,  but  A0  denotes  the  equivalent  conductance  at  infinite 
dilution. 

A  comparison  between  the  two  equations  is  afforded  by  the  fol- 
lowing table,  in  which  is  recorded  the  observed  and  calculated 
values  of  the  "  corrected  "  equivalent  conductance,  Arj/r}0)  for 
solutions  of  potassium  chloride,  at  18°. 

COMPARISON  OF  THE  EQUATIONS  OF  KRAUS  AND  BATES 


C 

*/% 

A  (obs.) 

Aif/fo 

AV/rio  (K.) 

An/%  (B.) 

3 

0.9954 

88.3 

87.89 

87.4 

89.3 

2 

0.9805 

92.53 

90.73 

90.9 

91.9 

1 

0.982 

98.22 

96.5 

96.4 

96.53 

0.5 

0.9898 

102.36 

101.32 

101.1 

101.29 

0.2 

0.9959 

107.90 

107.46 

107.6 

107.43 

0.1 

0.9982 

111.97 

111.77 

111.9 

111.73 

0.05 

0.9991 

115.69 

115.59 

115.5 

115.58 

0.02 

0.9996 

119.90 

119.85 

119.8 

119.83 

0.01 

0.9998 

122.37 

122.35 

122.4 

122.32 

0.005 

0.9999 

124.34 

124.33 

124  .4 

124.38 

0.002 

1.0000 

126.24. 

126.24 

126.3 

126.31 

0.001 

127.27 

127.27 

127.2 

127.32 

0.0005 



128.04 

,128.04 

127.6 

128.05 

0.0002 

128  .  70 

128  70 

127.9 

128.68 

0.0001 

129.00 

129^00 

128.1 

128.96 

0.0 

129.50 

100  ^o 

108  3 

100  Eft 

1  —  ./  .  *J\J 

j.^o  .  o 

J  —  ._/  .  «_HJ 

ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS         439 

It  will  be  observed,  that  for  dilute  solutions,  the  ratio,  r)/rj0,  is 
practically  unity,  and  furthermore,  that  the  value  of  CA/A0  is  so 
small,  that  the  second  term  on  the  right-hand  side  of  both  equations 
is  negligible  in  comparison  with  the  value  of  k. 

Therefore,  under  these  conditions,  both  equations  reduce  to  the 
form. 

A2C 

AO  (A0  -  A)  = 

which  will  be  recognized  as  identical  with  Ostwald's  dilution  law 
as  given  on  page  430. 

Heat  of  Ionization.  The  heat  of  ionization  of  an  electrolyte 
can  be  calculated  by  means  of  the  equation  of  the  reaction  isochore 
of  van't  Hoff  (see  p.  316),  provided  the  degree  of  ionization  at 
two  different  temperatures  is  known. 

Since  K\  =  -r— -  — y— , 

and 

K  az* 

(1 -<*)*' 

it  follows,  that  the  heat  of  ionization  may  be  calculated  by 
means  of  the  equation, 

2.3026  R  j  log      ^        -  log      J"\     |  TiT9 
T2-  Ti 

Arrhenius  *  has  shown,  that  this  equation  also  applies  to  those 
electrolytes  which  do  not  obey  the  Ostwald  dilution  law.  Some 
of  the  results  obtained  by  Arrhenius  are  given  in  the  following 
table. 

It  will  be  found,  that  the  values  of  the  heats  of  ionization  given 
in  this  table  do  not  agree  with  the  values  calculated  for  these 
same  substances  from  the  data  given  in  the  table  on  page  297. 
The  reason  for  this  lack  of  agreement  is,  that  the  data  of  the  earlier 
table  refer  to  the  heat  of  formation  of  the  ions  from  the  dissolved 
substance,  whereas  the  data  of  the  table  here  given,  represent  the 
combined  thermal  effects  of  solution  and  ionization. 

*  Zeit.  phys.  Chem.,  4,  96  (1889). 


440 


THEORETICAL  CHEMISTRY 


HEATS  OF  IONIZATION  OF  ELECTROLYTES 


Electrolyte. 

Temperature. 

Calories. 

Acetic  acid                                                    \ 

35° 

386 

Propionic  acid.                                               •] 

21°.  5 
35° 

—28 
557 

Butyric  acid                                                              •] 

21°.  5 
35° 

183 
935 

Phosphoric  acid.                     •] 

21°.  5 
35° 

427 
2458 

Hydrochloric  acid 

21°.  5 
35° 

210o 
1080 

Potassium  chloride     .    . 

35° 

362 

Potassium  bromide  ....           

35°    ' 

425 

Potassium  iodide  

35° 

916 

Sodium  chloride 

35° 

454 

Sodium  hydroxide 

35° 

1292 

Sodium  acetate. 

35° 

391 

The  Solubility  Product.  While  the  law  of  mass  action  does 
not  in  general,  apply  to  the  equilibrium  between  the  dissociated 
and  undissociated  portions  of  an  electrolyte,  —  except  in  the  case 
of  organic  acids  and  bases,  —  it  does  apply  with  a  fair  degree  of 
accuracy  to  saturated  solutions  of  electrolytes. 

A  saturated  solution  of  silver  chloride  affords  an  example  of 
such  an  equilibrium.  This  salt  is  practically  completely  ionized 
in  a  saturated  solution,  as  represented  by  the  equation, 

AgCl  <±  Ag  +  Cl'. 
Applying  the  law  of  mass  action  to  this  equilibrium,  we  obtain 

CAe-  X  Cc 


CAgCl 


K. 


Since  the  solution  is  saturated,  the  value  of  CAgci  must  remain 
constant  at  constant  temperature,  and  therefore, 

CAg-  X  ccr  =  constant  =  s,  (19) 

where  the  product  of  the  ionic  concentrations,  s,  is  called  the  solu- 
bility, or  ionic  product. 

The  equilibria  in  the  above  heterogeneous  system  may  be  repre- 
sented thus, 

Ag  +  Cl'  <=*  AgCl  <±  AgCl. 

(in  solution)         (solid) 

The  solubility  product  for  silver  chloride,  at  25°,  is  1.56  X  lO"10, 
the  ionic  concentrations  being  expressed  in  mols  per  liter.    Hence, 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS         441 

since  the  two  ions  are  present  in  equivalent  amounts,  a  saturated 
solution  of  silver  chloride,  at  25°,  must  contain  Vl.56  X  10~10 
=  1.25  X  10~5  mols  per  liter  of  Ag'  and  Cl'  ions.  In  general,  if 
the  equation 

nA  +±niAi  + 


represents  the  equilibrium  between  an  electrolyte  and  its  products 
of  dissociation  in  saturated  solution,  we  have 


The  solubility  product  may  be  defined  as  the  maximum  product  of 
the  ionic  concentrations  of  an  electrolyte  which  can  exist  in  equi- 
librium with  the  undissolved  phase  at  any  one  temperature. 

Just  as  the  dissociation  of  a  gaseous  substance,  or  of  an  organic 
acid,  is  depressed  by  the  addition  of  one  of  the  products  of  dis- 
sociation, so  when  a  substance  with  a  common  ion  is  added  to 
the  saturated  solution  of  an  electrolyte,  the  dissociation  is  de- 
pressed, and  the  undissociated  substance  is  precipitated. 

The  following  example  will  serve  to  illustrate  how  the  solu- 
bility product  of  a  substance  can  be  determined,  and  how  the 
change  in  solubility,  due  to  the  addition  of  a  substance  containing 
a  common  ion,  may  be  calculated.  The  solubility  of  silver  bromate, 
at  25°,  is  0.0081  mol  per  liter.  If  we  assume  complete  ionization, 
the  concentration  of  the  ions,  Ag*  and  Br(V,  will  be  identical 
and  equal  to  0.0081  mol  per  liter,  or 

(0.0081)  (0.0081)  =  s. 

The  solubility  of  silver  bromate  in  a  solution  of  silver  nitrate, 
containing  0.  1  mol  of  Ag*  ions,  can  be  calculated  from  the  equation, 

(0.0081)2  =  (0.0081  +  0.1  -  x)  (0.0081  -  x), 

where  x  represents  the  amount  of  silver  bromate  thrown  out  of  so- 
lution by  the  addition  of  0.1  mol  of  Ag-  ion.  Since  (0.0081  —  x) 
represents  the  concentration  of  BrO3'  ions,  after  the  addition  of 
the  silver  nitrate,  it  also  represents  the  solubility  of  silver  bromate 
under  similar  conditions.  The  effect  of  adding  a  solution  of  a 
soluble  bromate,  containing  0.1  mol  of  BrO3'  ion,  will  be  the  same 
as  that  produced  by  0.1  mol  of  Ag'  ion. 

The  Basicity  of  Organic  Acids.  The  Ostwald  dilution  law 
holds  strictly  for  all  monobasic  organic  acids,  and  also  for  poly- 


442 


THEORETICAL   CHEMISTRY 


basic  organic  acids,  which  are  less  than  50  per  cent  ionized.  The 
neutral  salts  of  these  acids,  however,  are  much  more  highly  ion- 
ized, and  the  difference  in  conductance  between  two  dilutions  of  a 
neutral  salt  of  a  polybasic  acid  is  greater  than  the  difference  in 
conductance  between  the  same  dilutions  of  a  neutral  salt  of  a 
monobasic  acid. 

Ostwald  *  has  shown,  that  it  is  possible  to  estimate  the  basicity 
of  an  organic  acid  from  the  difference  in  the  equivalent  conduct- 
ance of  its  sodium  salt  at  two  different  dilutions.  As  the 
result  of  a  long  series  of  experiments,  he  found  that  the 
difference  between  the  equivalent  conductance  of  the  sodium  salt 
of  a  monobasic  organic  acid,  at  v  =  32  liters,  and  at  v  =  1024  liters, 
is  approximately  10  units.  Similarly,  for  a  dibasic  acid,  the  differ- 
ence, between  the  same  dilutions  is  20  units,  and  for  an  n-basic 
acid  the  difference  is  10  n.  Hence,  to  estimate  the  basicity  of  an 
organic  acid,  the  equivalent  conductance  of  its  sodium  salt,  at  v  = 
32  liters,  and  at  v  =  1024  liters,  is  determined;  then,  if  A  is  the  dif- 
ference between  the  values  of  the  conductance  at  the  two  dilutions, 

the  basicity  will  be  n  =  —  • 

The  following  table  gives  the  values  of  A  and  n  for  the  sodium 
salts  of  several  typical  organic  acids. 

BASICITY  OF   ORGANIC  ACIDS 


Acid. 

A 

n 

Formic  

10.3 

1 

Acetic 

9  5 

1 

Propionic 

10  2 

1 

Benzoic                                              

8  3 

1 

Quininic                                          

19  8 

2 

Pyridine-tricarboxylic  (1,  2,  3)  .  

31  0 

3 

Pyridine-tricarboxylic  (1,  2,  4)  

29.4 

3 

Pyridine-tetracarboxylic  

41.8 

4 

Pyridine-pentacarboxylic 

50  1 

5 

Influence  of  Substitution  on  lonization.  Attention  has  already 
been  called  to  the  marked  difference  in  the  strength  of  acetic 
acid  produced  by  the  replacement  of  the  hydrogen  atoms  of  the 
methyl  group  by  chlorine.  In  the  accompanying  table  the  ioniza- 
tion  constants  for  various  substitution  products  of  acetic  acid 
are  given. 

*  Zeit.  phys.  Chem.,  i,  105  (1887);  2,  902  (1888). 


ELECTROLYTIC  EQUILIBRIUM   AND  HYDROLYSIS         443 
IONIZATION  CONSTANTS  OF  SUBSTITUTED  ACETIC  ACIDS 


Acid. 

lonization 
Constant  (25°). 

Acetic,  CH3COOH  

0  000018 

Propionic,  CH3CH2COOH  

0  000013 

Chloracetic,  CH2C1COOH  

0  00155 

Bromacetic,  CH2BrCOOH  

0  00138 

Cyanacetic,  CH2CNCOOH  

0  00370 

Glycollic.  CH2OHCOOH  

0  000152 

Phenylacetic,  C6H5CH2COOH  

0  000056 

Amidoacetic,  CH2NH2COOH  

3  4  X  10~10 

This  table  affords  an  interesting  illustration  of  the  influence  of 
different  substituents  on  the  strength  of  acetic  acid.  Thus,  the 
activity  of  the  acid  is  increased  by  the  replacement  of  alkyl  hydro- 
gen atoms  by  Cl,  Br,  CN,  OH,  or  C6H5,  while  the  substitution  of 
the  CH3,  or  NH2  groups  diminishes  its  activity.  If  we  assume 
that  the  substituents  retain  their  ion-forming  capacity  on  enter- 
ing into  the  molecule  of  acetic  acid,  these  differences  in  activity 
can  be  readily  explained.  Thus,  Cl,  Br,  CN,  and  OH  tend  to 
form  negative  ions,  and  hence  increase  the  negative  character  of 
the  group  into  which  they  enter.  On  the  other  hand,  basic  groups, 
such  as  NH2,  diminish  the  tendency  of  the  group  into  which  they 
enter  to  yield  negative  ions. 

The  influence  of  an  alkyl  residue  on  the  strength  of  an  organic 
acid  is  conditioned  by  its  distance  from  the  carboxyl  group.  This 
is  well  illustrated  by  the  ionization  constants  of  propionic  acid 
and  some  of  its  derivatives. 

IONIZATION  CONSTANTS  OF  PROPIONIC  ACID 
DERIVATIVES 


Acid. 

Ionization 
Constant  (25°). 

Propionic  acid   CH3CH2COOH                          

0.0000134 

Lactic  acid  CH3CHOHCOOH                     

0.000138 

/3-oxypropionic  acid  CH2OHCH2COOH 

0.0000311 

The  effect  of  the  OH  group,  in  the  a-position,  is  seen  to  be  much 
more  marked  than  when  it  occupies  the  /3-position. 

The  position  of  a  substituent  in  the  benzene  nucleus  exerts  a 
marked  influence  on  the  strength  of  the  derivatives  of  benzole 


444  THEORETICAL  CHEMISTRY 

acid.     The  ionization  constants  of  benzole  acid,  and  the  three 
chlorbenzoic  acids  are  given  in  the  following  table. 

IONIZATION  CONSTANTS  OF  BENZOIC  ACID 
DERIVATIVES 


Acid. 

Ionization  • 
Constant  (25°). 

Benzoic  acid,  C6H6COOH          

0  000073 

o-Chlorbenzoic  acid,  C6H4C1COOH  

0.00132 

m-Chlorbenzoic  acid  C6H4C1COOH 

0  000155 

p-Chlorbenzoic  acid  C6H4C1COOH.  ,  . 

0  000093 

When  the  halogen  enters  the  ortho-position,  the  strength  of  the 
acid  is  greatly  augmented,  while  in  the  meta-  and  para-positions, 
the  effect  is  much  smaller,  meta-chlorbenzoic  acid  being  stronger 
than  para-chlorbenzoic  acid.  It  is  a  general  rule,  that  the  influ- 
ence of  substituents  is  always  greatest  in  the  ortho-position,  and 
least  in  the  meta-  and  para-positions,  the  order  in  the  two  latter 
being  uncertain. 

Hydrolysis.  When  a  salt,  formed  by  a  weak  acid  and  a  strong 
base,  such  as  sodium  carbonate,  is  dissolved  in  water,  the  solution 
shows  an  alkaline  reaction.  On  the  other  hand,  when  a  salt, 
formed  by  a  strong  acid  and  a  weak  base,  such  as  ferric  chloride, 
is  dissolved  in  water,  the  solution  shows  an  acid  reaction. 

The  process  which  takes  place  in  the  aqueous  solution  of  a  salt, 
causing  it  to  react  alkaline  or  acid,  is  termed  hydrolysis,  or  hydro- 
lytic  dissociation.  If  MA  represents  a  salt,  in  which  M  is  the  basic, 
and  A  is  the  acidic  portion,  then  the  hydrolytic  equilibrium  may 
be  represented  by  the  equation, 

MA  +  H20  <F±  MOH  +  HA. 

If  the  base  formed  is  insoluble,  or  undissociated,  and  the  acid  is 
dissociated,  the  solution  will  react  acid.  If  the  acid  formed  is 
insoluble,  or  undissociated  and  the  base  is  dissociated,  the  solution 
will  react  alkaline.  Finally,  if  both  base  and  acid  are  insoluble, 
or  undissociated,  the  salt  will  be  completely  transformed  into  base 
and  acid,  and,  as  there  will  be  no  excess  of  either  H',  or  OH'  ions, 
the  solution  will  remain  neutral. 

It  is  evident  then,  that  hydrolysis  is  due  to  the  removal  of 
either  one,  or  both,  of  the  ions  of  water  by  the  ions  of  the  salt,  to 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS        445 

form  undissociated,  or  insoluble  substances.  As  fast  as  the  ions 
of  water  are  removed,  the  loss  is  made  good  by  the  dissociation  of 
more  water,  until  eventually,  a  condition  of  equilibrium  is  estab- 
lished. The  conditions  governing  hydrolytic  equilibrium  may  be 
determined  from  a  knowledge  of  the  solubility,  or  ionic  constant 
of  the  substances  involved.  Thus,  if  the  product  of  the  concen- 
trations of  the  ions,  M'  and  OH7,  exceeds  that  which  can  exist  in 
pure  water,  then  some  undissociated,  or  insoluble  substance  will 
be  formed.  This  will  disturb  the  equilibrium  of  H*  and  OH' 
ions,  and  a  further  dissociation  of  water  will  occur,  until  the 
ionic  product  of  water  is  just  reached.  If  the  ions  H*  and  A'  do 
not  unite  to  form  undissociated  acid,  the  presence  of  an  excess 
of  H*  ions  will  disturb  the  equilibrium  between  pure  water  and 
its  products  of  dissociation;  or,  since 

CH»  X 


the  concentration  of  OH'  ions  present,  will  be  -       where  CH-  rep- 

resents the  total  concentration  of  H*  ions.  A  similar  readjust- 
ment will  take  place  when  an  undissociated,  or  insoluble  acid, 
and  a  dissociated  base  are  formed. 

We  may  now  proceed  to  consider  three  different  cases  of  hydroly- 
sis, viz.,  when  the  reaction  is  caused  (1)  by  the  base,  (2)  by  the 
acid,  and  (3)  by  both  base  and  acid. 

CASE  I.  The  formation  of  an  undissociated,  or  insoluble  base  is  pri- 
marily the  cause  of  the  hydrolysis,  the  acid  formed  being  dissociated. 

Let  the  hydrolytic  equilibrium  be  represented  by  the  equation, 
MA  +  H20  ?=>  MOH  +  HA. 

The  reaction  will  proceed  in  the  direction  of  the  upper  arrow,  until 
the  product,  CM.  X  COH',  exceeds  that  which  can  exist  in  the  ab- 
sence of  an  undissociated  base.  When  equilibrium  is  established, 
we  have 

final  CM-  X  final  cOH'  =  #MOH  X  CMOH  formed,  (21) 

or,  if  the  base  formed  is  practically  insoluble,  the  equilibrium  equa- 
tion simplifies  to  the  form, 

final  CM-  X  final  COH'  =  SMOH,  (22) 

where  SMOH  is  the  solubility  product  of  the  base.     The  condition 
of  equilibrium  represented  by  the  equation, 
CH*  X  COH'  = 


446  THEORETICAL  CHEMISTRY 

must  be  fulfilled.  It  follows,  that  the  final  concentration  of  the 
OH'  ions  will  be  the  quotient  obtained  by  dividing  the  ionic 
product  for  water,  at  the  temperature  of  the  experiment,  by  the 
final  concentration  of  the  H*  ion,  this  latter  being  wholly  depend- 
ent upon  the  extent  of  the  reaction  and  the  degree  of  ionization  of 
the  acid  formed.  If  the  degree  of  hydrolysis  of  the  salt  be  repre- 
sented by  x,  and  the  degree  of  dissociation  of  the  unhydrolyzed 
portion  of  the  salt  be  denoted  by  as,  then,  if  one  mol  of  salt  be 
dissolved  in  V  liters  of  solution,  the  final  concentration  of  M*  ions 

will  be  as     y  —  -  ,  and  the  final  concentration  of  the  undissociated 

/y»  /y» 

base  will  be     .    The  total  acid  formed  will  be     ,  and  if  aa  denotes 


the  degree  of  dissociation  of  the  acid,  the  concentration  of  the  H* 
ions  will  be  aa 
(22),  we  obtain 


ions  will  be  aa     .     Substituting  these  values  in  equations  (21)  and 


Pis  (1  —  X)       SH20          ^  X 

y  —    =  A.MOH  X   y  > 

**Y 

and 

a,  (1  -  x)     sH2o 

-y-  ----  -  =  SMOH-  (24) 


Simplifying  equations  (23)  and  (24),  we  have 

xz          aa  _    SH?O        ~ 
(1  -  x)  V  •  5T~  K^  -  K*  (25) 


and 

x         aa        sH,o        v , 

:  *» .  (26) 


From  equations  (25)  and  (26)  it  appears  that  the  constant  of  hy- 
drolysis can  be  found  either  from  the  ionic  product  for  water  and 
the  ionization  constant  of  the  base,  or  from  the  ionic  product  for 
water  and  the  solubility  product  of  the  base.  Furthermore,  if  the 
base  formed  is  insoluble,  equation  (26),  shows  that  the  degree  of 
hydrolysis,  x,  is  independent  of  the  dilution  of  the  salt,  V. 

CASE  II.  The  formation  of  an  undissociated  or  insoluble  acid 
is  primarily  the  cause  of  the  hydrolysis,  the  base  formed  being  dis- 
sociated. In  this  case,  hydrolysis  takes  place  until  the  product 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS        447 

cH-  X  cA'  exceeds  that  which  can  exist  in  the  absence  of  undis- 
sociated  acid.     When  equilibrium  is  established,  we  have 

final  cH-  X  final  cA'  =  KHA  X  CHA  formed,  (27) 

or,  if  the  acid  formed  is  practically  insoluble,  the  equilibrium  equa- 
tion simplifies  to  the  form, 

final  cH-  X  final  cA'  =  SHA.  (28) 

Since  the  final   CH*  =  sH2o  -r-    final   COH',  we   have,   final  CA>  = 

^y  -  ,  final  COH'  =  <xi>  -y  ,  where  ab  is  the  degree  of  dissocia- 

tion of  the  base  formed,  and  the  final  CHA  =  -^  .     Substituting 
these  values  in  equations  (27)  and  (28),  we  obtain 

<*«  (1  -  X)       SR20          ~  X 

and 


y  X  y  y 


aa  (1  —  x)     SH,O  /Qnx 

-y -=SHA.  (30) 


Simplifying  equations  (29)  and  (30),  we  have 
&  «6       ^H2o       r 

a———-      .      -    =53  -      533     ^^ 

-  x)  V     aa      AHA 
and 


It  is  evident  from  equations  (31)  and  (32),  that  the  constant  of 
hydrolysis  can  be  found  either  from  the  ionic  product  for  water  and 
the  ionization  constant  of  the  acid,  or  from  the  ionic  product  for 
water  and  the  solubility  product  of  the  acid. 

CASE  III.  The  formation  of  an  add  and  a  base,  both  being 
slightly  dissociated,  is  the  cause  of  the  hydrolysis. 

In  this  case  let  us  assume  that  ^HA  is  smaller  than  KMOH. 

Since  the  final  cOH'  =  KM°H  XCMOH,  and  since  both  HA  and  MOH 

CM- 

x 
are  slightly  dissociated,  we  may  write  CHA  ==  CMOH  =  y,  and 

CA,  =  CM.  .  -.(IP*). 


448  THEORETICAL  CHEMISTRY 

Substituting  these  values  in  equation  (27),  we  obtain 

as  (1  -  x)         SHZO  ^  x 

~y  ----  J-  =  *HA  X  r  (33) 

-K-MOH 


Simplifying  equation  (33),  we  obtain 

' 


(1  -  X?  «82 

From  equation  (34)  we  see  that  the  constant  of  hydrolysis  can  be 
found  from  the  ionic  product  for  water  and  the  ionization  constants 
of  the  acid  and  the  base.  If  both  acid  and  base  are  practically 
insoluble,  the  reaction  will  be  complete  at  all  dilutions. 

As  an  illustration  of  the  application  of  the  foregoing  equations, 
we  may  take  the  calculation  of  the  degree  of  hydrolytic  dissociation 
of  potassium  cyanide  in  0.1  molar  solution,  at  25°.  Potassium 
cyanide  being  a  salt  of  a  weak  acid,  the  degree  of  hydrolysis  can 
be  calculated  by  means  of  the  equation, 


(1  -  x}  V 


as 


Since  at  25°  C.,  #HA  =  7.2  X  1Q-10  and  sH2o  =  (1.05  X  10~7)2,  we 
have 

K    =  SH.O       (1.05  X  IP"7)2 
XHA          7.2  X  10-10    : 

and  since  in  dilute  solution  as  =  ab  =  1,  we  have 

x2         =  (1.05  X  I0~7f 
(1  -  a;)  10         7.2  X  10-10 
or 

x  =  0.0123. 

Experimental  Determination  of  Hydrolysis.  The  degree  of 
hydrolysis  can  be  determined  experimentally  in  several  different 
ways.  A  very  convenient  method  is  that  based  upon  measure- 
ments of  electrical  conductance.  When  a  salt  reacts  hydroly  tic- 
ally  with  one  mol  of  water,  the  limiting  value  of  its  equivalent 
conductance  will  be  AA  +  As,  where  AA  and  AB  denote  the  equiv- 
alent conductances  of  the  acid  and  base  formed.  If  A  is  the  equiv- 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS        449 

alent  conductance  of  the  unhydrolyzed  salt,  and  Aft  is  the  actual 
conductance  of  the  salt  at  the  same  dilution,  then  the  increase  in 
conductance,  corresponding  to  a  degree  of  hydrolysis,  x,  will  be 
Aft  -  A.  The  value  of  A  may  be  found  by  determining  the 
conductance  of  the  salt  in  the  presence  of  an  excess  of  one  of  the 
products  of  hydrolysis,  and  deducting  from  it  the  conductance 
of  the  substance  added.  If  the  hydrolysis  were  complete,  the 
equivalent  conductance  would  be  AA  +  AB  —  A,  hence  we  have 

«-A  i*:A  ..  (35) 

AA  H-  AB  —  A 

all  conductances  being  measured  at  the  same  dilution  and  at  the 
same  temperature.  The  following  example  will  illustrate  the 
use  of  this  equation :  —  At  25°,  the  equivalent  conductance  of  an 
aqueous  solution  of  aniline  hydrocfhloride  is  118.6,  the  dilution 
being  99.2  liters.  The  equivalent  conductance,  in  the  presence  of 
an  excess  of  aniline,  is  103.6,  while  the  equivalent  conductance  of 
hydrochloric  acid,  at  the  same  dilution,  is  381.  The  conductance 
of  pure  aniline  is  so  small  as  to  be  negligible.  Substituting  these 
values  in  equation  (35),  we  find 

118.6  -  103.6 
=   381  -  103.6 

Lunden  *  has  shown  how  this  method  may  be  extended  to  cases 
where  both  acid  and  base  are  slightly  dissociated. 

In  addition  to  the  foregoing  method,  the  degree  of  hydrolysis 
may  also  be  determined  by  measuring  the  catalytic  effect  of  the 
hydrogen,  or  hydroxyl  ion  on  the  rate  of  inversion  of  sugar,  f  or  on 
the  rate  of  decomposition  of  diazoacetic  ester.  { 

Another  method,  of  limited  applicability,  to  which  we  shall 
refer  in  the  following  chapter,  involves  the  calculation  of  the 
hydrogen  ion  concentration  in  a  partially  hydrolyzed  solution, 
from  measurements  of  electromotive  force.  § 

The  values  of  the  degree  of  hydrolysis  of  various  salts,  at  25°, 
in  0.1  normal  solutions  are  given  in  the  accompanying  table. 

*  Jour.  chim.  phys.,  5,  145,  574  (1907). 

t  Ley,  Zeit.  phys.  Chem.,  30,  222  (1899). 

|  Bredig  and  Fraenkel,  Zeit.  Elektrochem.,  n,  525  (1905). 

§  Denham,  Jour.  Chem.  Soc.,  93,  41  (1908). 


450 


THEORETICAL  CHEMISTRY 


DEGREE  OF  HYDROLYSIS  OF  SALTS   AT  25°* 

HYDROLYSIS  OF  H YD ROCHLO RIDES  OF  WEAK  BASES 

(Measured  by  catalytic  decomposition  of  esters) 


Base 

Percentage 
Hydrolysis 

Base 

Percentage 
Hydrolysis 

Glycocoll 

19 

Urea  

90 

Asparagine 

25 

Acetamide  

98 

Acetoxime 

36 

Thiourea 

99 

HYDROLYSIS  OF  ALKALI  SALTS  OF  WEAK  ACIDS 
(Measured  by  catalytic  saponification  of  esters) 


Acid 

Percentage 
Hydrolysis 

Acid 

Percentage 
Hydrolysis 

Hydrocyanic 

1.12 

p-Chlorphenol  

1.62 

Acetic 

0.008 

p-Cyanphenol  

0.29 

Carbonic  
Phenol  

3.17 
3.05 

p-Nitrophenol  

0.16 

HYDROLYSIS  OF  HYDROCHLORIDES  OF  WEAK  ORGANIC  BASES 
(Measured  by  rate  of  inversion  of  cane  sugar) 


Base 

Percentage 
Hydrolysis 

Base 

Percentage 
Hydrolysis 

Pyridine     

1.2 

Urea  

81 

Quinoline  

1.2 

Acetamide  

78 

Aniline  

2.6 

Asparagine  

21 

HYDROLYSIS  OF  METALLIC  CHLORIDES 
(Measured  by  rate  of  inversion  of  cane  sugar) 


Base 

Percentage 
Hydrolysis 

Base 

Percentage 
Hydrolysis 

Zinc 

0  1  at  100° 

Aluminium 

2  7  at    77° 

Lead 

0  2  at  100° 

Cerium 

0  3  at  100° 

Beryllium.  . 

1  8  at  100° 

Lanthanum 

0  1  at  100° 

Aluminium  

6.1  at  100° 

Iron  (ferric)  .  . 

10    at    40° 

*  Farmer,  Report  British  Association,  240,  (1901). 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS        451 

The  lonization  Constant  of  Water.  One  of  the  most  accurate 
methods  known  for  the  determination  of  the  ionization  constant 
of  water,  is  based  upon  measurements  of  the  degree  of  hydrolytic 
dissociation  of  different  salts.  Thus,  Shields  *  found  that  a  0.1 
molar  solution  of  sodium  acetate  is  0.008  per  cent  hydrolyzed  at 
25°.  We  may  consider  the  salt,  as  well  as  the  sodium  hydroxide 
formed  from  its  hydrolysis,  to  be  completely  dissociated  at  this 
dilution.  The  ionization  constant  of  the  acetic  acid  formed  is 
0.000018,  at  25°.  Solving  equation  (31)  for  SH,O,  and  remember- 
ing that  as  =  otb  =  1,  we  have 


Substituting  the  above  values  in  this  expression,  we  obtain 


SH,o  =  0.000018  •  1Q  =  1.16  X  10- 


and  since  the  ions,  H*  and  OH',  are  present  in  equivalent  amounts 
we  have 

CH-  =  COH'  =  Vl.16  X  10~14  =  1.1  X  10-7mol  per  liter. 


Kohlrausch  obtained  from  his  measurements  of  the  conductance 
of  pure  water  at  25°,  CH*  =  COH'  =  1.05  X  10~7  mol  per  liter  (see 
p.  421). 

The  ionization  constant  of  water  has  been  measured  by  several 
other  independent  methods,  the  agreement  between  the  results 
being  quite  satisfactory.  The  different  methods  employed  may 
be  classified  as  follows: 

(1)  Measurement  of  the  conductance  of  water  of  a  high  degree 
of  purity, 

(2)  Measurement  of  the  hydrolysis  of  salts, 

(3)  Measurement  of  the  catalytic  effects  of  H*  and  OH'  on  the 
saponification  of  methyl  acetate, 

(4)  Measurement  electromotive  force,  (see  next  chapter). 

The  following  table  gives  a  summary  of  the  various  measure- 
ments of  the  ionization  constant  of  water. 

*  Zeit.  phys.  Chein.,  12,  167  (1893). 


452 


THEORETICAL  CHEMISTRY 


IONIZATION  CONSTANT  OF  WATER 


Tempera- 
ture 

0° 
18 
25 
50 
75 
100 

CH'  =  COH'  in  gram-mols  per  liter  X  107 

I* 

nt 

IIIJ 

IV  § 

v|| 

vi  H 

VII** 

12'.  3 

0.298 
0.68 
0.90 
2.1 
4.1 
6.9 

0.37 

~':Ll6 
2.96 
5.3 

8.6 

i!o3 

2.28 

'i!i' 

'i'.i) 

0.88 
1.03 



*  Noyes  and  Kanolt,  Carnegie  Inst.  Publ.,  63,  346  (1907).  Hydrolysis 
of  ammonium  diketotetrahydrothiazole. 

t  Lorenz  and  Bohi,  Zeit.  phys.  Chem.,  66,  733  (1909).  Electromotive 
force  of  acid-alkali  cell. 

J  Kohlrausch,  Wied.  Ann.,  53,  209  (1894).     Conductance  of  pure  water. 

§  Lunden,  Jour.  chim.  phys.,  5,  574  (1907).  Hydrolysis  of  ammonium 
borate. 

||  Shields,  Zeit.  phys.  Chem.,  12,  167  (1893).  Hydrolysis  of  sodium 
acetate. 

If  Hudson,  Jour.  Am.  Chem.  Soc.,  29,  1571  (1907).  Mutarotation  of 
glucose  by  acids  and  alkalies. 

**  Beans  &  Oakes,  Jour.  Am.  Chem.  Soc.,  42,  2116  (1920).  Electro- 
motive force  of  cell,  |  Hg  |  HgCl,KCl  |  KC1  |  H2O  hH2. 

Theory  of  Indicators.  When  a  solution  of  a  base  is  titrated 
with  an  acid,  the  end-poyit  is  reached  when  the  amount  of  acid 
added  is  just  equivalent  to  the  amount  of  base  present.  Theo- 
retically, the  solution  at  the  end  of  the  titration  should  be  identical 
in  composition  with  a  solution  prepared  by  dissolving  a  known 
weight  of  the  salt  formed  in  the  process  of  neutralization,  in  the 

calculated  amount  of  the  solvent,  xUnless  both  acid  and  base 
u  -j  4.'  i  •  •  A-  i**fi  '^so^f  *#ffw&£6es  s* 

have  identical  lomzation   constants, A  however,   the   salt  formea 

will  be  hydrolyzed,  and,  therefore,  the  indicator  should  change 
its  color  when  the  hydrogen  ion  concentration  of  the  solution 
acquires  the  same  value  as  that  which  exists  in  a  solution  of  the 
salt  formed  in  the  process  of  neutralization. 

In  order  to  understand  the  mechanism  of  the  action  of  indi- 
cators, it  must  be  remembered  that  indicators  are,  in  almost  all 
cases,  weak  organic  acids  which  are  capable  of  existing  in  two 
tautomeric  forms  in  equilibrium  with  each  other.  For  example, 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS        453 

phenolphthalein  has  been  shown  by  Rosensteia  *  to  exist  in  the 
two  following  forms: 

XC6H4OH  /CeH4O,  H* 

r*  TI       r*/  r1  TT       r\s 

U6±14  — U<  i       Wtl4  —  U<T 

|  |   \C5H4OH      and  \C6H4  =  O 

CO    -O  COO,  H* 

One  of  these  forms,  which  for  convenience  we  may  designate  by 
the  formula,  HIn,  is  a  non-electrolyte,  while  the  other  form,  which 
may  be  represented  by  the  formula,  HIn',  is  an  electrolyte  and, 
as  such,  is  ionized  in  solution.  The  equilibria  existing  in  a  solution 
of  such  an  indicator-acid  may  be  represented  in  the  following 
manner:  %t 

HIn  *=>  HIn'  <=±  H'  +  In', 
(color  A)  (color  B) 

The  criteria  of  a  good  indicator  are,  that  there  shall  be  a  sharp 
contrast  between  the  two  colors  which  it  exhibits  in  acid  and 
alkaline  media,  and  that  the  value  of  ratio,  HIn '/HIn,  shall  be 
small.  It  is  apparent  that  the  addition  of  a  strong  acid  to  a  solu- 
tion containing  an  indicator-acid  will  displace  the  above  equilibria 
toward  the  left,  with  the  production  of  the  so-called  "  acid  color  " 
of  the  indicator.  In  like  manner,  the  addition  of  a  base  to  the 
solution  of  an  indicator-acid  will  displace  the  equilibria  toward 
the  right,  with  the  development  of  the  so-called  "  alkaline  color  " 
of  the  indicator.  The  "  neutral  color  "  of  an  indicator  is  inter- 
mediate between  the  acid  and  alkaline  colors,  and  is  obtained 
when  the  concentration  of  the  hydrogen  ion  in  the  solution  assumes 
such  a  value,  that  one-half  of  the  indicator  is  in  the  form  HIn,  and 
the  other  half  is  in  the  form  In'  and  MIn,  M  being  the  metal  of 
the  added  base. 

According  to  the  law  of  mass  action, 

CH'  X  Cln/  =  Ka  (36) 

Cnin 

or 

•  TT    Cnin 


Cm' 


(37) 


If  we  denote  by  y  the  fraction  of  the  indicator  which  exists  in  the 
form  exhibiting  the  alkaline  color,  then  1  -  y  will  represent  the 

*  Jour.  Am.  Chem.  Soc.,  34,  117  (1912). 


454  THEORETICAL  CHEMISTRY 

portion  which  is  present  in  the  form  exhibiting  the  acid  color. 
Therefore,  equation  (37)  becomes 

(1  -  y) 


If  y  =  0.5,  the  indicator  will  exhibit  its  neutral  color  and  equation 
(38)  will  take  the  form 

CH-  =  Ktt.  (39) 

In  other  words,  when  the  indicator  acquires  its  neutral  color,  the 
indicator  constant  Ka,  is  numerically  equal  to  the  hydrogen  ion 
concentration  of  the  solution.  The  value  of  the  fraction  y,  for 
different  indicator|jjnas  been  determined  by  measuring  the  num- 
ber of  cubic  centimeters  of  a  solution  of  a  given  base  which  must  be 
added  to  a  solution  containing  a  known  amount  of  indicator  to 
bring  about  a  change  of  color.*  Thus,  it  has  been  found  that 
phenolphthalein  changes  color  when  y  =  0.1.  The  value  of  the 
indicator  constant  Ka,  for  phenolphthalein,  being  1  X  10~10,  it 
follows  that  the  hydrogen  ion  concentration,  at  which  the  color 
change  occurs,  will  be  ^^ 

cH-  =  1  X  10-lo(1  ~®'V  =  1  X  10~9. 

Therefore,  phenolphthalein  will  be  a  suitable  indicator  to  use  in 
titrations  where  the  hydrogen  ion  concentration  of  the  solution* 
of  the  resulting  salt  is  approximately  1  X  10~9.     For  example,^ 
when  acetic  acid  is  titrated  with  sodium  hydroxide,  the  proper 
end-point  will  be  reached,  when  the  hydrogen  ion  concentration 
existing  in  the  solution  corresponds  to  that  of  a  solution  of  sodium 
acetate  of   equivalent   concentration.     Since   sodium   acetate   is 
the  salt  of  a  weak  acid,  it  follows,  that  its  degree  of  hydrolysis 
can  be  calculated  by  means  of  equation  (31).     Substituting  the 
known  data  in  this  equation,  we  have 

a2  1  X  IP"14 

(l-x)V.    18  X  10-6' 

it  being  assumed  that  the  solution  is  so  dilute  that  as  =  ab  =  1. 
If  the  concentration  of  the  sodium  acetate  at  the  end-point  is  0.1 
normal,  the  degree  of  hydrolysis  will  be  7  X  10~5,  and  the  hydroxyl 

*  See  Salm,  Biochem.  Zeit.,  21,  131  (1909);  also  "  Theory  of  Indicators," 
by  Prideaux. 


ELECTROLYTIC  EQUILIBRIUM   AND  HYDROLYSIS        455 


ion  concentration  will  be  7  X  10~°.  Since  CH-  X  COH'  =  1  X  10~14, 
it  follows,  that  the  hydrogen  ion  concentration  in  the  solution  will 
be  1  X  10-14  5-  7  X  10-6  •  1.4  X  1Q-9.  That  is,  the  solution  is 
actually  slightly  alkaline  at  the  end-point,  and  consequently  phenol- 
phthalein,  which  undergoes  a  change  in  color  at  practically  the 
same  hydrogen  ion  concentration,  is  a  proper  indicator  to  employ 
in  this  titration. 

REFERENCES 

Chemical  Statics  and  Dynamics.     Mellor.     Chapter  IX. 

A  System  of  Physical  Chemistry.     Lewis.     Vol.  I,  Chapter  V. 

Theory  of  Indicators.     Prideaux. 

PROBLEMS 

1.  At  25°  the  specific  conductance  of  butyric  acid  at  a  dilution  of  64 
liters  is  1.812  X  10~4  reciprocal  ohms.     The  equivalent  conductance  at 
infinite  dilution  is  380  reciprocal  ohms.     What  is  the  degree  of  ioniza- 
tion  and  the  concentration  of  H*  ions  in  the  solution?    What  is  the  ioni- 
zation  constant  of  the  acid? 

Ans.  a  =  0.0305,  CH-  =  4.765  X  10~4  mol  per  liter,  K  =  1.5  X  10-5.' 

2.  The  heat  of  neutralization  of  nitric  acid  by  sodium  hydroxide  is 
13,680  calories,  and  of  dichloracetic  acid,   14,830  calories.     When  one 
equivalent  of  sodium  hydroxide  is  added  to  a  dilute  solution  containing 
one  equivalent  of  nitric  acid  and  one  equivalent  of  dichloracetic  acid, 
13,960  calories  are  liberated.     What  is  the  ratio  of  the  strengths  of  the 
two  acids? 

3.  For  potassium  acetate  we  have  the  following  data:  — 


X 

A  (18°) 

2 

67.1 

10 

78.4 

100 

87.9 

10000 

91.9 

and  ZOQ   =  64.67,  and  l^   =  35.     Compare  the  constants  obtained  by  the 
K'  CH3COO' 

Ostwald,  Rudolphi,  and  van't  Hoff  dilution  laws. 

4.  The  ionization  constant  of  a  0.05  molar  solution  of  acetic  acid  is 
9.0000175  at  18°,  and  0.00001624  at  52°.  Calculate  the  heat  of  ioniza- 
tion of  the  acid.  To  what  temperature  does  this  value  correspond? 

Ans.  416  cal.  at  35°. 


456  THEORETICAL  CHEMISTRY 

/"  5.  At  20°  the  specific  conductance  of  a  saturated  solution  of  silver 
bromide  was  1.576  X  10~~G  reciprocal  ohms,  and  that  of  the  water  use<. 
was  1.519  X  10~8  reciprocal  ohms.  Assuming  that  silver  bromide  is 
completely  ionized,  calculate  the  solubility  and  the  solubility  product  of 
silver  bromide,  having  given  that  the  equivalent  conductances  of  potas- 
sium bromide,  potassium  nitrate,  and  silver  nitrate  at  infinite  dilution 
are  137.4,  131.3,  and  121  reciprocal  ohms  respectively. 

Am.  CAgBr  =  449  X  10~7  mol  per  liter. 

6.  The  solubility  of  silver  cyanate  at  100°  is  0.008  mol  per  liter.  Cal- 
culate the  solubility  in  solution  of  potassium  cyanate  containing  0.1  mol 
of  K*  ions. 

•  \[)  Calculate  the  degree  of  hydrolytic  dissociation  of  a  0.1  molar  solu- 
tion of  ammonium  chloride,  having  given  the  following  data:  —  as  = 
0.86,  aa  =  0.87,  #NH4OH  =  0.000023,  and  sn2o  =  (0.91  X  lO"7)*  at  25°. 

Ans.  x  =  0.006  per  cent. 

8.   In  the  reaction  represented  by  the  equation 

MA3  +  3  H20  =  M  (OH),  +  3  HA, 
the  base  formed  is  insoluble.     Derive  an  expression  for  the  constant  of 

S3H2O  (3z)3         eta3 

hydrolysis.  Ans.  Kv  = —  =  j~, — ~  -  —  • 

SM'OH),        (!-*)«•      as 

/9.  The  equivalent  conductance  of  aniline  hydrochloride  at  a  dilution 
of  197.6  liters  is  126.7  reciprocal  ohms,  at  25°.  The  equivalent  con- 
ductance of  aniline  hydrochloride  in  the  presence  of  an  excess  of  aniline 
is  106.6;  and  the  equivalent  conductance  of  hydrochloric  acid  at  the 
same  dilution  is  415.  If  the  conductance  of  pure  aniline  is  negligible, 
calculate  the  degree  of  hydrolytic  dissociation  and  the  constant  of  hydrol- 
ysis, assuming  as  =  aa  =1. 

Ans.  x  =  6.52  per  cent,  Ki  =  2.30  X  1Q-5. 

10.  The  hydrolysis  constant  of  aniline  is  2.25  X  1Q-5,  and  the  ioniza- 
tion  constant  is  5.3  X  lO"10.  Calculate  the  concentration  of  the  H* 
and  OH'  ions  in  water.  Am.  CH-  =  COH'  =  1.09  X  10~7. 

The  value  of  the  indicator  constant  for  methyl  orange  is  Ka  =  5  X 
and  the  indicator  changes  to  its  alkaline  color  when  it  is  nine-tenths 
neutralized.  Show  that  methyl  orange  is  a  suitable  indicator  to  use  in 
the  titration  of  ammonium  hydroxide  with  hydrochloric  acid.  The 
ionization  constant  of  ammonium  hydroxide  may  be  assumed  to  be  18  X 
1CH. 

12.  The  value  of  the  indicator  constant  for  rosolic  acid  is  Ka  =  4  X 
10~8  and  the  indicator  changes  to  its  alkaline  color  when  it  is  one-tenth 
neutralized.  Would  this  be  a  suitable  indicator  to  use  in  the  preceding 
titration? 


CHAPTER  XVII 
ELECTROMOTIVE  FORCE 

Galvanic  Cells.  Since  the  year  1800,  when  Volta  invented 
his  electric  pile,  many  different  forms  of  galvanic  cell  have  been 
introduced.  It  is  not  our  purpose  to  give  a  detailed  account  of 
these  cells,  but  rather  to  give  a  brief  outline  of  the  theories  which 
have  been  advanced  in  explanation  of  the  electromotive  force 
developed  in  such  cells.  When  two  metallic  electrodes  are  immersed 
in  a  solution  of  an  electrolyte,  a  current  will  flow  through  a  wire 
connecting  the  electrodes,  provided  the  two  metals  are  dissimilar. 
Not  only  can  an  electric  current  be  obtained  from  a  combination 
of  two  different  metals  in  the  same  electrolyte,  but  also  from  two 
different  metals  in  two  different  electrolytes,  from  the  same 
metal  in  different  electrolytes,  or  from  the  same  metal  in  two 
different  concentrations  of  the  same  electrolyte.  In  order 
that  the  electromotive  force  of  the  combination  shall  remain 
constant,  it  is  necessary  that  the  chemical  changes  involved 
in  the  production  of  the  current  shall  neither  destroy  the  differ- 
ence between  the  electrodes,  nor  deposit  upon  either  of  them  a 
non-conducting  substance.  A  galvanic  combination  which  ful- 
fils these  conditions  very  satisfactorily  is  the  Daniell  cell.  This 
cell  consists  of  zinc  and  copper  electrodes  immersed  in  solutions  of 
their  salts,  as  represented  by  the  scheme, 


Zn  - 


-  Sol.  of  ZnS04  1|  Sol.  of  CuSO4  -  Cu, 

in  which  the  two  vertical  lines  indicate  a  porous  partition  separat- 
ing the  two  solutions.  When  the  zinc  and  copper  electrodes  are 
connected  by  a  wire,  a  current  of  positive  electricity  passes,  from 
the  copper  to  the  zinc,  along  the  wire.  Zinc  dissolves  from  the 
zinc  electrode,  an  equivalent  amount  of  copper  being  displaced 
from  the.  solution  and  deposited  simultaneously  on  the  copper 
electrode.  As  long  as  only  a  moderate  current  flows  through  the 
cell,  the  original  nature  of  the  electrodes  is  not  modified,  the  only 
change  which  occurs  being  the  gradual  dilution  of  the  copper  sul- 

457 


458  THEORETICAL  CHEMISTRY 

phate,  owing  to  the  separation  of  copper  and  its  replacement  by 
zinc.  If  the  loss  of  copper  sulphate  is  replaced,  the  electromotive 
force  of  the  cell  will  remain  constant.  If,  after  the  cell  is  assembled, 
the  circuit  remains  open,  the  copper  sulphate  will  slowly 
diffuse  into  the  solution  of  zinc  sulphate,  and  metallic  copper  will 
ultimately  be  deposited  on  the  zinc  electrode.  In  this  way  mini- 
ature, local  galvanic  cells  will  be  formed  on  the  surface  of  the  zinc, 
and  the  metal  will  dissolve  as  though  the  main  circuit  were  closed. 
Until  this  deposition  takes  place,  the  cell  may  be  left  on  open 
circuit  without  danger  of  deterioration.  Unless  chemically  pure 
zinc  is  used,  local  action  is  likely  to  occur,  owing  to  the  formation  of 
local  galvanic  couples  between  the  impurities  in  the  electrode,  — 
chiefly  iron,  —  and  the  zinc.  This  action  may  be  prevented  by 
amalgamating  the  zinc  electrode.  In  this  process  the  mercury 
dissolves  the  zinc  and  not  the  iron,  a  uniform  surface  of  the  former 
metal  being  produced. 

An  interesting  experiment,  due  to  Ostwald,  *  illustrates  the  con- 
ditions essential  to  the  continuous  production  of  an  electric  current. 
Two  electrodes,  one  of  amalgamated  zinc  and  the  other  of  platinum, 
are  each  immersed  in  a  solution  of  potassium  sulphate,  the  two 
solutions  being  separated  by  a  porous  cup.  When  the  two  elec- 
trodes are  connected  by  means  of  a  wire,  no  permanent  flow  of 
current  occurs.  An  inappreciable  quantity  of  zinc,  however,  goes 
into  solution,  since  any  current  must  necessarily  first  liberate 
potassium  at  the  platinum  electrode.  The  potassium  thus  set  free 
reacts  with  the  water.  This  process  requires  the  expenditure  of 
more  energy  than  the  solution  of  the  zinc  supplies.  If  sulphuric 
acid  is  added  to  the  compartment  containing  the  zinc,  the  con- 
dition of  the  system  will  be  unchanged;  i.'e.,  the  zinc  will  remain 
undissolved.  If,  on  the  other  hand,  a  few  drops  of  sulphuric  acid 
are  added  to  the  compartment  containing  the  platinum  electrode, 
bubbles  of  hydrogen  will  appear,  and  the  zinc  will  dissolve  with 
the  simultaneous  development  of  an  electric  current.  This  experi- 
ment shows,  that  in  order  that  positively  charged  ions  may  enter 
a  solution,  an  equivalent  amount  of  negatively  charged  ions  must 
be  introduced,  or  an  equivalent  amount  of  positively  charged 
ions  must  be  removed. 

Reversible  Cells.  Galvanic  cells  are  either  reversible,  or  ir- 
reversibk,  according  as  the  processes  taking  place  within  them  can 
*  Phil.  Mag.  [5],  32,  145  (1891). 


ELECTROMOTIVE    FORCE  459 

be  reversed  or  not.  If  we  disregard  the  slow  processes  of  diffusion, 
the  Daniell  cell  may  be  taken  as  an  example  of  an  almost  perfect 
reversible  element.  If  an  electromotive  force  slightly  less  than 
that  of  the  cell  be  applied  to  it,  in  the  reverse  direction,  the  current 
within  the  cell  will  flow  from  the  zinc  to  the  copper  electrode,  as 
usual.  On  the  other  hand,  if  the  external  electromotive  force 
slightly  exceeds  that  of  the  cell,  the  current  within  the  cell  will 
flow  in  the  reverse  direction,  zinc  being  deposited  and  copper  dis- 
solved. Any  cell  from  which  gas  is  evolved  is  irreversible,  since  the 
passage  of  a  current  in  the  reverse  direction  cannot  restore  the 
cell  to  its  original  condition. 

Relation  between  Chemical  Energy  and  Electrical  Energy. 
Helmholtz  and  Thomson  were  the  first  to  propose  a  satisfactory 
theory  of  the  action  of  the  reversible  cell.  According  to  this  theory, 
the  energy  of  the  chemical  process  taking  place  within  the  cell  was 
considered  to  be  completely  transformed  into  electrical  energy.  It 
was  soon  shown  that  this  theory  is  inadequate,  since,  with  the  ex- 
ception of  the  Daniell  cell,  the  chemical  energy  is  not  equivalent 
to  the  electrical  energy  produced. 

The  exact  quantitative  relationship  between  the  chemical 
energy  transformed,  and  the  maximum  electrical  energy  devel- 
oped, by  a  reversible  galvanic  cell  can  readily  be  derived  by  means 
of  the  familiar  Gibbs-Helmholtz  equation  (see  p.  139), 

(i) 

where  W  denotes  the  maximum  work  obtainable  from  a  com- 
pletely reversible,  isothermal  process,  and  where  U  is  the  decrease 
in  the  total  energy  of  the  system.  If  the  maximum  work  be 
expressed  in  terms  of  electrical  energy,  we  will  have 

W  =  nFE,  (2) 

in  which  F  is  the  faraday,  E,  the  electromotive  force  of  the  cell 
and  n,  the  number  of  unit  charges  transferred.  On  differentiating 
equation  (2),  we  obtain 

dW  =  nFdE,  (3) 

and  on  substituting  these  values  of  W  and  dW  in  equation  (1) 
we  have 

nFE  -  U  = 


460 


or 


THEORETICAL  CHEMISTRY 
U 


(4) 


Since  the  decrease  in  internal  energy  is  equal  to  the  heat  evolved, 
at  constant  volume,  equation  (4)  may  be  written  in  the  form, 

E=^Q^+T^dE\  (5) 

This  equation  expresses  the  electromotive  force  of  a  reversible  cell 
in  terms  of  the  temperature  coefficient  of  the  cell,  and  the  heat  of 
the  chemical  reaction  occurring  within  the  cell. 

In  the  following  table,  compiled  by  Jahn,*  the  values  of  the 
heat  of  the  reaction  several  different  cells  are  given,  (a)  as  cal- 
culated by  the  Gibbs-Helmholtz  equation,  and  (b)  as  calculated 
by  the  Helmholtz-Thomson  rule. 

EXPERIMENTAL  VERIFICATION  OF  GIBBS- 
HELMHOLTZ   EQUATION 


Cell 

E 

dE/dT 

U  (calc.) 

U  (obs.) 

nFE/J 

Cu|Cu(C2H302)2 
solution 

|Pb(C2H302)2, 

100  H2O|Pb 

(0°) 

0.4764 

+0.000385 

16,900 

17,533 

21,684 

Ag|AgCl,ZnCl2, 
100H2O|Zn 
(0°) 

1.015 

-0.000402 

51,989 

52,046 

46,907 

Hg|HgO  n  NaOH|H2 

(18°) 

0.9243 

-0.00031 

46,750 

46,700 

42,590 

Hg|HgCL0.01  nKCl| 
nKNO3|0.01  n- 
KOH|Hg20|Hg 

(18.5°) 

0.1656 

+0.000387 

-3,710 

-3,280 

7,566 

It  will  be  seen  that  the  agreement  between  the  observed  and 
calculated  values  of  U  is  as  close  as  could  be  expected,  whereas, 
the  figures  given  in  the  last  .column,  calculated  according  to  the 
Helmholtz-Thomson  rule,  differ  widely  from  the  experimentally 

determined  values.     When  j=,  =  0,  E  becomes.equal  to  r-|p  ;  in 
*  Wied.  Ann.,  28,  21  (1886). 


ELECTROMOTIVE    FORCE  461 

other  words,  when  the  temperature  coefficient  of  the  cell  is  zero, 
the  electrical  energy  is  equal  to  the  chemical  energy.  This  is  true 
of  the  Daniell  cell,  which  has  an  extremely  small  temperature  co- 
efficient. 

For  cells  in  which  the  electromotive  force  varies  appreciably 
with  the  temperature,  it  is  possible  to  calculate  the  value  of  the 
electromotive  force  at  any  temperature  by  means  of  the  Gibbs- 
Helmholtz  equation,  provided  the  temperature  coefficient  is  known. 
In  the  Grove  gas  cell,*  E  =  1.062  and  Qv  =  34,200  calories,  hence 

TdE  -  1 062  -  34'200  -        0418 

1dT~  96,540  X  0.2394  ~ 

The  value  determined  by  direct  experiment  is  —  0.416  volt.  The 
Gibbs-Helmholtz  equation  shows,  that  the  amount  of  heat  accom- 
panying a  chemical  process  does  not  alone  furnish  a  measure  of 
the  electrical  energy  which  may  be  obtained  from  it,  since  the 
heat  which  is  absorbed  from  the  surrounding  medium  may  also 
be  transformed  into  electrical  energy,  or  the  output  of  electrical 
energy  may  be  less  than  the  heat  of  the  reaction  within  the  cell. 

Solution  Pressure.  It  is  a  familiar  fact,  that  water  has  a 
tendency  to  assume  the  form  of  vapor,  and  if  the  vapor  be  contin- 
ually removed  from  its  surface,  a  definite  mass  of  water  will  grad- 
ually be  completely  transformed  into  the  state  of  vapor.  The 
pressure  of  the  vapor,  at  any  one  temperature,  is  a  measure  of  the 
tendency  of  water  to  undergo  this  transformation.  This  tendency 
of  water  to  assume  a  form  other  than  that  in  which  it  actually 
exists,  is  typical  of  all  substances.  Attention  has  already  been 
directed  to  this  fact  in  connection  with  the  application  of  the  law 
of  mass  action  to  heterogeneous  equilibria.  It  was  then  pointed 
out,  that  all  solids  have  a  definite  vapor  pressure  at  a  definite 
temperature,  which  is  independent  of  the  amount  of  solid  present. 
When  a  solid,  such  as  cane  sugar,  is  brought  in  contact  with  water, 
it  tends  to  pass  into  solution.  This  tendency  is  constant,  at 
constant  temperature,  since  the  active  mass  of  the  solid  is  constant. 
From  the  close  analogy  between  the  vapor  state  and  the  dissolved 
state,  the  tendency  of  a  solid  to  pass  into  solution  is  termed  the 
solution  pressure.  A  dissolved  solid,  on  the  other  hand,  also  shows 
a  tendency  to  separate  from  the  solution  as  the  concentration  is 

*  The  Grove  gas  cell  may  be  here  represented  by  the  scheme:  Pto2  -  acid- 
ulated water  —  Ptn2. 


462  THEORETICAL  CHEMISTRY 

increased.  When  the  solution  becomes  supersaturated,  the  tend- 
ency of  the  solute  to  separate  in  the  solid  form  is  greater  than 
the  tendency  of  the  solid  to  dissolve.  It  is  evident,  from  these 
considerations,  that  the  pressure  exerted  by  the  dissolved  solid 
is  its  osmotic  pressure,  and  whether  the  solid  will  dissolve,  or 
separate  from  the  solution  depends  upon  whether  the  solution 
pressure  is  greater,  or  less  than  the  osmotic  pressure. 

This  conception  of  solution  pressure  was  introduced  by  Nernst,* 
and  in  conjunction  with  the  theory  of  electrolytic  dissociation,  it 
has  proved  of  great  value  in  affording  a  much  deeper  insight  into 
the  mechanism  of  the  development  of  differences  in  potential 
within  a  galvanic  cell.  Thus,  when  a  metal  is  dipped  into  water 
it  tends  to  dissolve  owing  to  its  solution  pressure,  P,  and,  in  con- 
sequence of  this  tendency,  it  sends  a  certain  number  of  positive 
ions  into  solution.  The  solution  thus  becomes  positively  charged, 
and  the  metal,  which  was  initially  neutral,  acquires  a  negative 
charge,  due  to  the  loss  of  a  certain  amount  of  positive  electricity. 
This  process  will  cease  when  the  solution  becomes  so  strongly 
charged  with  positive  electricity  that  it  prevents  the  separation 
of  any  more  positive  ions  from  the  metal.  Relatively  few  ions 
leave  the  metal  before  equilibrium  is  established,  since  the  charge 
on  each  ion  is  so  great;  in  fact,  the  concentration  of  metal  ions  in 
the  solution  is  much  too  small  to  be  detected  analytically.  When 
a  metal  is  dipped  into  a  solution  of  one  of  its  salts,  however,  the 
conditions  are  altered.  In  this  case,  the  positive  ions  of  the  metal, 
already  present  in  the  solution,  oppose  the  entrance  of  more  posi- 
tive ions,  and  the  equilibrium  between  these  two  opposing 
tendencies  will  be  conditioned  by  the  relative  values  of  the  solu- 
tion pressure,  P,  of  the  metal,  and  the  osmotic  pressure,  p,  of  the 
ions  of  the  dissolved  salt. 

It  is  evident  that  the  three  following  conditions  are  possible: 

(1)  If  P  >  p,  the  metal  will  continue  to  send  ions  into  the 
solution  until  the  accumulated  charges  in  the  solution  oppose 
further  action.     The  solution  acquires  a  positive  charge,  and  the 
metal  a  negative  charge. 

(2)  If  P  <  p,  the  positive  ions  of  the  dissolved  salt  will  sepa- 
rate on  the  metal  until  the  accumulated  charges  oppose  further 
action.     The  metal  acquires  a  positive  charge  and  the  solution  a 
negative  charge. 

*  Zeit.  phys.  Chem.,  4,  150  (1889). 


ELECTROMOTIVE  -FORCE 


463 


(3)  If  P  =  p,  no  action  will  take  place  and  no  difference  of 
potential  will  be  established  between  the  metal  and  the  solution. 
These  three  cases  are  represented  diagrammatically  in  Fig.  114. 
When  equilibrium  is  established  and  the  metal  is  negative  against 
the  solution,  the  metal  is  surrounded  by  a  layer  of  positively 
charged  ions.  This  constitutes  what  is  known  as  a  Helmholtz 
electrical  double  layer.  If  positive  electricity  be  communicated  to 


Fig.  114 

the  metal,  the  double  layer  will  be  broken,  and  more  ions  will 
pass  from  the  metal  into  the  solution,  but  as  soon  as  the  supply 
of  positive  electricity  is  cut  off,  the  double  layer  will  again  be 
formed.  Similarly,  when  the  metal  is  positive  against  the  solu- 
tion, an  electrical  double  layer  will  be  formed,  the  metal  being 
surrounded  by  a  layer  of  negatively  charged  ions. 

The  actual  existence  of  a  Helmholtz  double  layer  has  been 
demonstrated  by  Palmaer.*  In  his  experiments,  Palmaer  allowed 
exceedingly  minute  globules  of  mercury  to  fall  into  a  dilute  solu- 
tion of  mercurous  nitrate,  contained  in  a  tall  vessel,  the  bottom 
of  which  was  covered  with  a  layer  of  pure  mercury,  as  shown  in 
Fig.  115.  Since  the  solution  pressure  of  mercury  is  less  than  the 
osmotic  pressure  of  the  Hg*  ions,  each  drop  of  mercury,  as  it 
enters  the  solution,  will  acquire  a  positive  charge,  and  if  the  theory 
of  the  electrical  double  layer  is  correct,  this  positively  charged 
globule  should  attract  negatively  charged  ions  and  drag  them  down 
through  the  solution.  When  the  globule  reaches  the  mercury,  at 

Zeit.  phys.  Chem.,  25,  265  (1898);   28,  257  (1899);  36,  664  (1901). 


464 


THEORETICAL  CHEMISTRY 


the  bottom  of  the  vessel,  it  will  give  up  its  positive  charge,  and  as 
many  Hg*  ions  will  pass  into  solution  as  there  are  N(V  ions  in  the 
double  layer.  The  solution  will  thus  become  more  concentrated 
just  above  the  layer  of  mercury  on  the  bottom  of  the  vessel.  Pal- 

maer's  experiments  showed 
that  this  difference  in  con- 
centration is  actually  pro- 
duced, in  some  cases  the 
concentration  in  the  upper 
part  of  the  solution  being 
reduced  as  much  as  50  per 
cent. 

The  metals  sodium,  potas- 
sium, .  .  .  zinc,  cadmium, 
cobalt,  nickel,  and  iron  are 
negative  against  solutions  of 
their  salts;  i.  e.,  P  >  p.  The 
noble  metals  are  generally 
positive  against  solutions  of 
their  salts,  or  P  <  p.  The 
anions  are,  so  far  as  is  known, 
positive  to  solutions  of  their 

salts.     Electrolytic    solution 

pressure  varies  with  the  tem- 
perature, with  the  nature  of  the  solvent,  and  also  with  the  con- 
centration of  the  active  substance  in  the  electrode. 

The  Difference  of  Potential  between  a  Metal  and  a  Solution. 
From  the  foregoing  considerations,  it  is  possible  to  derive  an 
equation  expressing  the  difference  of  potential  between  a  metallic 
electrode  and  a  solution  of  one  of  its  salts. 

Let  us  imagine  one  gram-ion  of  a  metal  to  be  transferred  from 
the  electrolytic  solution  pressure  P,  to  the  osmotic  pressure  p. 
The  osmotic  work  done  will  be 


f 

JP 


(6) 


Integrating  this  expression,  we  have 


Osmotic  work 


(7) 


ELECTROMOTIVE    FORCE  465 

The  corresponding  electrical  energy  gained  is  nFE,  where  E  is  the 
difference  of  potential  between  the  metal  and  the  solution,  F  =  1 
faraday  =  96,500  coulombs,  and  n  is  the  valence  of  the  metal. 
Since  the  osmotic  work  done  is  equivalent  to  the  electrical  energy 
gained,  we  may  equate  these  two  expressions,  as  follows: 

nFE  =  RT  \oge-, 
or 


Expressing  both  sides  of  equation  (8)  in  electrical  units,  and  trans- 
forming to  Briggsian  logarithms,  we  obtain 

2  P 

E  =  96,540  X  n  X^O.4343  X  0.2394/  log  p  ' 
or 

js.^riogf  (9) 

For  univalent  ions  at  17°,  we  have 

E  =  0.0575  log--  (10) 

In  a  galvanic  cell  composed  of  two  metals,  each  immersed  in  a 
solution  of  one  of  its  salts,  a  difference  of  potential  may  be  estab- 
lished (1)  at  the  junction  of  two  metals,  (2)  at  the  junction  of 
the  two  solutions,  and  (3)  at  the  points  of  contact  of  the  metals 
with  their  respective  solutions.  If  the  temperature  remains  con- 
stant, (1)  is  negligible,  and  in  general,  (2)  is  exceedingly  small; 
therefore,  the  electromotive  force  of  the  cell  may  be  considered  as 
due  to  tl^e  differences  of  potential  arising  at  the  two  electrodes* 
Assuming  the  temperature  to  be  17°*  the  electromotive  force*  of 
the  cell  will  be 


The  Measurement  of  Electromotive  Force.  The  value  of  the 
electromotive  force  of  a  cell  may  vary  with  the  conditions  of  meas- 
urement. Since,  according  to  Ohm's  law,  E  =  /  (R  +  r),  where 
R  is  the  resistance  of  the  external  circuit,  and  r  is  the  internal 
resistance  of  the  cell,  it  follows  that  the  fall  of  potential,  IR,  in 


466  THEORETICAL  CHEMISTRY 

the  external  circuit  will  only  be  equal  to  E,  when  r  is  negligible 
in  comparison  with  R.  Furthermore,  when  the  circuit  is  closed, 
the  electrodes  of  the  cell  frequently  become  polarized,  owing  to 
the  deposition  of  the  products  of  electrolysis,  and  an  opposing 
electromotive  force  is  set  up. 

To  avoid  these  difficulties,  the  electromotive  force  is  usually 
measured  on  open  circuit  by  the  Poggendorff  compensation 
method.  In  this  method,  the  electromotive  force  to  be  measured 
is  just  balanced  by  an  equal  and  opposite  electromotive  force,  so 
that  no  current  passes.  The  arrangement  of  th»  apparatus  for 
such  measurements  is  shown  in  Fig.  116.  If  the  two  ends  of  the 


Fig.  116 

wire,  AB,  of  a  Wheatstone  bridge  are  connected  to  a  lead  accumu- 
lator, C,  there  will  be  a  uniform  fall  of  potential  along  its  entire 
length.  The  amount  of  fall,  along  any  portion  AD,  will  be  pro- 

AD 
portional  to  the  length  AD,  and  equal  to  the  fraction   --r-^  of 


the  total  fall  of  potential  along  the  entire  length  of  the  wire.  Now 
let  one  terminal  of  a  cell,  whose  electromotive  force  is  less  than 
that  of  C,  be  connected  to  A,  and  the  other  terminal  be  connected, 
through  a  galvanometer  G,  with  a  sliding  contact  D,  the  two  cells  E 
and  C  working  in  opposition.  A  current  will  flow  through  the 
circuit  AEGD,  and  will  be  indicated  by  the  galvanometer  at  all 
positions,  except  that  at  which  the  fall  of  potential  along  the  wire, 
from  A  to  D,  is  equal  to  the  electromotive  force  of  E.  Hence  we 
have 

e.m.f.  of  C  :  e.m.f.  of  E  ::  AB  :  AD, 


ELECTROMOTIVE    FORCE  467 

from  which  the  value  of  the  electromotive  force  of  the  cell  E,  can 
be  calculated.  Since  the  electromotive  force  of  a  lead  accumu- 
lator is  not  quite  constant,  it  is  customary,  after  having  deter- 
mined the  point  D,  to  substitute  a  standard  cell  for  E,  and  balance 
this  against  the  accumulator,  finding  a  new  point  of  balance  D'. 
We  now  have  the  proportion, 

e.m.f.  of  C  :  e.m.f.  of  standard  ::  AB  :  AD'. 
Combining  these  two  proportions,  we  obtain 

e.m.f.  of  E  :  e.m.f.  of  standard  ::  AD  :  ADf. 

Instead  of  using  a  galvanometer,  as  a  "  null "  instrument  for  indi- 
cating when  the  point  of  balance  has  been  reached,  a  capillary 
electrometer  is  ordinarily  employed. 

In  precise  determinations  of  electromotive  force,  the  simple 
slidewire  bridge  and  capillary  electrometer  is  usually  replaced  by 
a  potentiometer  and  sensitive  galvanometer,  thereby  greatly  in- 
creasing the  accuracy  of  the  measurements.  Recently,  Beans 
and  Oakes  *  have  developed  a  method,  whereby  the  electromotive 
force  of  cells  having  very  high  internal  resistance  can  be  measured 
with  an  accuracy  of  0.5  millivolt.  In  this  method,  the  cell  is 
connected  to  a  condenser  of  suitable  capacity,  and  when  a  suffi- 
cient quantity  of  electricity  has  accumulated,  the  condenser  is  dis- 
charged through  a  calibrated  ballistic  galvanometer.  In  addition 
to  its  accuracy,  this  method  involves  the  use  of  much  less  expensive 
apparatus  than  is  required  in  the  more  elaborate  potentiometer 
method. 

Standard  Cells.  It  is  apparent,  that  the  accuracy  of  all  meas- 
urements of  electromotive  force  is  dependent  upon  the  cell  em- 
ployed as  a  standard.  Much  time  has  been  devoted  to  the 
study  of  various  reversible  elements,  with  a  view  to  establishing  a 
standard  of  electromotive  force.  As  a  result,  we  have  the  com- 
plete specifications  for  two  standard  cells,  either  of  which  may  be 
readily  reproduced. 

(a)  The  Weston,  or  Cadmium  Standard  Cell.  The  most  widely 
used  standard  of  electromotive  force  is  the  so-called  Weston  cell, 
made  up  according  to  the  scheme, 

Hg  -  Solution  Hg2S04 1|  Solution  CdS04  -  Cd. 
*  Jour.  Am.  Chem.  Soc.,  42,  2116  (1920). 


468  THEORETICAL  CHEMISTRY 

A  short  platinum  wire  is  sealed  through  the  bottom  of  each  limb 
of  an  H-shaped  vessel.  In  one  limb  is  placed  a  small  amount  of 

a  10  to  15  per  cent  cadmium  amalgam,  together  with  a  layer  of 

g 
small  crystals  of  CdSO4-  ~H2O.     In  the  other  limb  is  placed  a 

o 

small  amount  of  pure  mercury,  over  which  is  a  layer  of  a  paste, 
composed  of  solid  mercurous  sulphate  and  a  saturated  solution 
of  cadmium  sulphate.  The  cell  is  then  filled  with  crystals  of 
cadmium  sulphate  and  a  saturated  solution  of  cadmium  sulphate, 
after  which  the  two  limbs  of  the  cell  are  hermetically  sealed.  If 
carefully  prepared,  this  cell  will  remain  unaltered  for  years  and 
will  have  an  electromotive  force,  at  20°,  of  1.0183  volts.  In  addition 
to  the  fact  that  it  can  be  so  easily  reproduced,  the  temperature 
coefficient  of  the  cell  is  almost  negligible. 

The  electromotive  force  of  a  Weston  standard  cell  at  any  tem- 
perature t,  is  given  by  the  formula, 

e.m.f.  at  t°  =  1.0183  -  0.000038  (t  -  20).  (12) 

(b)  The  Clark,  or  Zinc  Standard  Cell.  Until  several  years 
ago,  the  Clark  cell  was  considered  to  be  the  most  trustworthy 
standard  of  electromotive  force.  This  cell  is  made  up  according 
to  the  scheme, 

Hg  -  Solution  Hg2S04  ||  Solution  ZnS04  -  Zn. 

The  construction  of  the  cell  is  similar  to  that  of  the  Weston  cell. 
It  may  be  reproduced  with  as  great  accuracy  and  with  no  more 
trouble  than  the  Weston  cell,  but  its  relatively  large  temperature 
coefficient  renders  it  less  satisfactory.  The  electromotive  force 
of  the  Clark  standard  cell  at  any  temperature  t,  may  be  calculated 
by  means  of  the  formula, 

e.m.f.  at  t°  =  1.4328  -  0.00119  (t  -  15)  -  0.00007  (t  -  15)2.  (13) 

The  Capillary  Electrometer.  When  pure  mercury  is  covered 
with  sulphuric  acid,  its  surface  tension  is  diminished.  This  may 
be  shown  by  the  following  experiment:  In  a  small  evaporating 
dish  place  about  5  cc.  of  pure  mercury,  and  cover  it  with  a  10  per 
cent  solution  of  sulphuric  acid  to  which  has  been  added  enough 
potassium  dichromate  to  impart  a  light  yellow  color  to  the  so- 
lution. The  globule  of  mercury  will  immediately  flatten  out, 
indicating  that  its  surface  tension  has  diminished.  If  now  the 
mercury  be  touched  with  a  piece  of  iron  wire,  it  will  instantly 


ELECTROMOTIVE  FORCE 


469 


contract,  until  the  contact  with  the  wire  is  broken;  it  will  then 
flatten  out,  until  it  again  comes  in  contact  with  the  wire,  when  the 
globule  of  mercury  will  once  more  contract.  In  this  way,  a  regular 
pulsation  of  the  mercury  may  be  obtained. 

This  interesting  phenomenon  was  observed  early  in  the  nineteenth 
century  by  Henry,  but  was  first  satisfactorily  explained  by  Lipp- 
mann  *  in  1873.  Lippmann  showed,  that  when  the  globule  of  mer- 
cury is  negatively  electrified,  its  surface  tension  increases  and  the 
drop  shrinks.  If  sufficient  negative  electricity  is  imparted  to  the 
mercury,  it  is  possible  to  restore  the  globule  to  its  original  form. 
On  applying  more  negative  electricity,  the  globule  of  mercury  again 
expands.  When  the  iron  wire  touches  the  globule  it  charges  it 
negatively,  because  when  the  iron  dissolves,  it  furnishes  positively 
charged  ions  to  the  solution,  and  thus  acquires  a  negative  charge 
which  it  imparts  to  the  mercury.  At  the  same  time,  the  chromic 
acid  in  the  solution  undergoes  reduction  to  chromium  sulphate. 
Lippmann  concluded  from  his  experiments, 
that  the  difference  of  potential  arises  at  the 
surface  of  contact  between  the  mercury  and 
the  solution  of  the  electrolyte,  and  that  the 
surface  tension  of  the  mercury  is  a  function 
of  the  difference  of  potential.  Making  use 
of  this  principle,  he  constructed  the  capillary 
electrometer,  a  convenient  form  of  which  is 
shown  in  Fig.  117.  The  bulb  A,  through  the 
bottom  of  which  a  platinum  wire  is  sealed, 
contains  pure  mercury  and  dilute  sulphuric 
acid  (1  :  6).  Pure  mercury  is  poured  into 
the  other  limb  of  the  electrometer  until 
it  stands  at  B  in  the  wide  tube,  and 
at  C  in  the  capillary  tube.  Owing  to 
the  capillary  depression  of  the  mercury,  C 
lies  below  B.  Electrical  connection  with  the 
mercury  is  established  at  B,  by  means  of  a 
platinum  wire.  The  position  of  the  mercury  in  the  capillary  is 
determined  by  its  surface  tension;  if  the  surface  tension  is  in- 
creased, the  mercury  will  descend;  if  it  is  diminished,  the  mercury 
will  ascend.  If  a  negative  charge  is  communicated  to  the  mercury 
at  B,  the  surface  tension  will  be  increased  and  the  meniscus  will 
*  Pogg,  Ann.,  149,  546  (1873). 


117 


470 


THEORETICAL  CHEMISTRY 


descend;  if  a  positive  charge  is  imparted  to  the  mercury  at  B,  the 
surface  tension  will  be  diminished  and  the  meniscus  will  ascend. 
The  amplitude  of  the  movement  of  the  meniscus  is  an  inverse 

function  of  the  diameter  of  the  capil- 
lary tube.  If  the  meniscus  be  ob- 
served through  a  microscope  provided 
with  an  eye-piece  micrometer,  as 
shown  in  Fig.  118,  it  is  possible  to 
detect  very  slight  movements,  and  to 
measure  differences  of  potential  less 
than  0.0001  volt.  The  capillary  elec- 
trometer is  an  excellent  "  null  " 
instrument. 

In  using  the  electrometer  no  large 
electromotive  force  should  be  applied, 
since  the  meniscus  surface  becomes 
polarized  very  easily.  If  this  should 
occur,  a  new  surface  may  be  secured 
by  blowing  gently  at  B  and  forc- 
ing a  drop  of  mercury  out  of  the 
capillary  into  the  bulb.  Lippmann 
studied  the  effect  of  steadily  increasing 
potentials  on  the  movement  of  the  meniscus.  Pitting  move- 
ments of  the  meniscus  on  the  axis  of 
ordinates,  and  potentials  on  the  axis 
of  abscissae,  he  found  that  there  is  a 
maximum  in  the  curve  corresponding 
to  about  0.8  volt.  This  is  the  elec- 
tromotive force  which  must  be  applied, 
in  order  to  counterbalance  the  difference 
of  potential  produced  by  the  contact  of 
dilute  sulphuric  acid  with  the  surface 
of  the  mercury.  At  the  meniscus 
surface,  an  electrical  double  layer  is 
formed.  The  mercury  is  positively 
charged,  and  above  it  there  must  be 
a  layer  of  negatively  charged  ions,  as 
shown  in  Fig.  119.  Just  how  this  double  layer  is  formed  is  not 
known  with  certainty,  but  it  has  been  suggested  that  the  slight 
film  of  oxide,  which  is  probably  present  on  the  surface  of  the 


Fig.   118 


+    -f    -f- 


Sulphuric  Acid- 


Mercury- 


Fig.    119 


ELECTROMOTIVE  FORCE 


471 


is     ~ 


purest  mercury,  dissolves  in  the  sulphuric  acid  forming  a  solution 
of  mercurous  sulphate,  and  from  this  solution,  the  positively 
charged  Hg*  ions  deposit  on  the  mercury,  giving  it  a  positive 
charge.  Whether  this  explanation  is  correct  or  not,  the  fact  re- 
mains, that  the  mercury  is  positive  against  the  solution. 
;  Normal  Electrodes.  The  method  commonly  employed  for 
the  measurement  of  the  difference  of  potential  between  a  metal 
and  a  solution,  is  based 
upon  the  use  of  an  elec- 
trode in  which  the  differ-  Ml^ 
ence  of  potential  between 
the  electrode  and  a  certain 
solution  of  one  of  its  salts 
is  known.  Such  an  elec- 
trode is  called  a  normal 
electrode.  If  a  cell 
made  up  by  combining 
the  normal  electrode  with 
the  electrode  whose  po- 
tential is  to  be  determined, 
it  is  possible,  from  meas- 
urements of  the  resulting 
electromotive  force,  to 
calculate  the  value  of 
the  unknown  difference 
of  potential.  The  most 
convenient  electrode  to  prepare,  is  the  normal  calomel  elec- 
trode, a  satisfactory  form  of  which  is  shown  in  Fig.  120.  The 
bottom  of  the  electrode  vessel  is  covered  with  a  layer  of  pure 
mercury,  upon  which  is  poured  a  paste,  prepared  by  rubbing  to- 
gether in  a  mortar  mercury  and  calomel,  moistened  with  a  molar 
solution  of  potassium  chloride.  The  vessel  is  then  filled  with  a 
molar  solution  of  the  same  salt  which  has  been  saturated  with 
calomel  by  prolonged  shaking  with  the  latter.  Connection  with 
the.  mercury  is  established  by  means  of  a  platinum  wire  sealed  into 
a  glass  tube  A,  the  latter  being  passed  through  the  rubber  stopper 
which  closes  the  vessel.  In  using  the  calomel  electrode,  the  bent 
side  tube,  C,  is  filled  with  molar  potassium  chloride  by  applying 
suction  at  the  side  tube  B,  which  is  then  closed  by  means  of  a 
pinch-cock. 


w 


Hgr+HgCl 
Hgr 


Fig.  120 


472 


THEORETICAL  CHEMISTRY 


The  difference  of  potential,  at  any  temperature  t,  of  the  calo- 
mel electrode  prepared  as  described,  and  represented  by  the 
scheme, 

Hg  -  Solution  HgCl  in  molar  KC1, 

is 

E  =  +  0.56  { 1  +  0.0006  (t  -  18) }  volt.  (14) 

The  positive  sign  before  the  value  0.56,  indicates  that  the  electrode 
is  positive  to  the  solution.  In  order  to  measure  the  potential  of 
another  electrode  by  means  of  the  calomel  electrode,  the  arrange- 
ment shown  in  Fig.  121  is  com- 
monly used.  Here,  A  rep- 
resents the  "  half  -element  "  of 
which  the  potential  is  to  be 
determined,  B  represents  the 
side  tube  of  the  calomel  elec- 
trode, and  C  represents  an 
intermediate,  connecting  vessel 
containing  a  molar  solution  of 
potassium  chloride.  In  cases 
where  potassium  chloride 
forms  a  precipitate  with  the 
electrolyte  in  A,  the  solution 

in  C  may  be  replaced  by  a  molar  solution  of  potassium  nitrate, 
without  altering  the  value  of  the  electromotive  force  of  the  cell. 
The  original  measurement  of  the  potential  of  the  calomel 
electrode  was  made  by  forming  a  cell  with  this  and  another 
electrode  whose  potential  against  its  solution  is  zero.  Such  an 
electrode  is  known  as  a  null  electrode.  Thus,  if  a  copper 
electrode  is  immersed  in  a  solution  of  copper  sulphate,  the 
Cu"  ions  will  leave  the  solution  and  charge  the  electrode 
positively.  If  now  a  solution  of  potassium  cyanide  is  added, 
the  nearly  undissociated  salt,  K2Cu2  (CN)4,  will  be  formed, 
and  by  adding  a  sufficient  amount  of  the  solution,  the  concen- 
tration of  the  Cu"  ions  may  be  reduced  until  the  metal  and  the 
solution  have  the  same  potential.  The  addition  of  more  potassium 
cyanide  will  still  further  diminish  the  osmotic  pressure  of  the  Cu" 
ions,  and  the  electrode  will  acquire  a  negative  charge.  Similarly, 
mercury  in  a  solution  of  a  double  cyanide  may  be  used  as  a  null 
electrode. 
Another  form  of  null  electrode  is  the  so-called  dropping  elec- 


ELECTROMOTIVE  FORCE  473 

trade  of  Helmholtz.*  The  principle  involved  in  this  electrode 
has  already  been  discussed  in  connection  with  Palmaer's  experi- 
ment (p.  463).  An  extremely  fine  stream  of  mercury  is  allowed 
to  flow  from  a  funnel  having  a  minute  capillary  orifice,  the  stem  of 
the  funnel  dipping  below  the  surface  of  a  molar  solution  of  potas- 
sium chloride  containing  mercurous  ions.  As  each  little  globule 
enters  the  solution,  it  acquires  a  positive  charge  and  attracts  the 
negatively  charged  ions  of  the  electrolyte,  dragging  them  down 
with  itself.  When  each  globule  reaches  the  layer  of  mercury  at 
the  bottom  of  the  vessel,  its  surface  and  capacity  are  diminished, 
and  as  many  Hg*  ions  leave  the  layer  of  mercury  and  enter  the 
solution  as  there  were  negatively  charged  ions  carried  down  by 
the  globule.  This  process  continues  until  the  osmotic  pressure 
of  the  remaining  ions  is  equal  to  the  solution  pressure  of  the  metal. 
The  mercury,  both  in  the  stream  and  at  the  bottom  of  the  vessel, 
has  the  same  potential  as  the  solution.  If  now,  the  difference  of 
potential  between  the  mercury  in  the  funnel  and  the  mercury  in 
the  vessel  be  measured,  we  shall  obtain  the  potential  of  mercury 
against  a  molar  solution  of  potassium  chloride. 

The  dropping  electrode  was  for  a  long  time  regarded  as  an 
ideal  standard  of  potential,  but  Nernst  has  pointed  out  a 
number  of  serious  objections  to  it.  Until  a  wholly  satisfac- 
tory standard  of  potential  is  obtained,  he  proposes  that  the 
potential  of  the  hydrogen  electrode  be  adopted  as  the  standard. 
This  consists  of  a  strip  of  platinized  platinum,  half  being 
immersed  in  pure  hydrogen  gas,  and  half  in  a  solution  of 
sulphuric  acid  of  such  concentration  that  it  shall  contain  1 
gram  of  hydrogen  ions  per  liter.  The  use  of  the  hydrogen 
electrode  as  a  standard  is  purely  arbitrary,  but  there  are 
many  advantages  in  referring  differences  of  potential  to  ^  this 
standard.  Owing  to  certain  experimental  difficulties  attending 
the  use  of  this  electrode,  it  is  customary  to  make  the  actual 
measurements  with  the  calomel  electrode,  and  then  refer  them 
to  the  hydrogen  standard,  taking  the  potential  of  the  calomel 
electrode  to  be  +  0.283  volt,  when  referred  to  the  hydrogen 
electrode  as  zero.  The  positive  sign  indicates  that  the  electrode 
is  positive  towards  the  solution. 

Measurement  of  the  Difference  of  Potential  between  a  Metal 
and  a  Solution.  The  difference  of  potential  between  a  metal 
*  Ann.  der  Phys.,  44,  42  (1890). 


474  THEORETICAL  CHEMISTRY 

and  a  solution  of  one  of  its  salts  is  easily  determined  by  means  of 
the  calomel  electrode.  For  example,  in  order  to  determine  the 
potential  of  zinc  against  a  molar  solution  of  zinc  sulphate,  the 
electromotive  -jorce  E,  of  the  combination, 

Zn  -  m  ZnSO4  1|  m  KC1,  HgCl  -  Hg, 

(cal.  electrode) 

is  measured,  and  found  to  be  1.08  volts,  the  mercury  being  the 
positive  terminal  of  the  cell.  Applying  the  Nernst  equation, 


in  which  PI  and  pi  denote  the  solution  pressure  and  the  osmotic 
pressure  of  the  zinc  ions,  and  P2  and  p2  denote  the  solution  pressure 
and  the  osmotic  pressure  of  the  mercury  ions,  we  have 


1.08  =  0.56  -log 
or 


£i  =  0.56  -  1.08  =  -  0.52  volt. 


That  is,  the  zinc  electrode  is  negative  against  a  molar  solution  of 
zinc  sulphate,  the  difference  of  potential  being  0.52'  volt.  As  an 
example  of  a  cell  in  which  the  mercury  of  the  calomel  electrode 
is  the  negative  terminal  of  the  cell,  we  may  take  the  following 
combination  :  — 

Cu  -  m  CuSO4  |[  m  KC1,  HgCl  -  Hg. 

The  electromotive  force  of  this  cell  is  0.025  volt.  Since  the 
current  flows  from  the  copper  to  the  mercury,  we  have. 

D/TT  p 

E  =  0.025  =  |±logeg-  0.56, 
or 

p  m  p 

±\oge£l  =  0.025  +  0.56  =  0.585  volt. 


That  is,  the  copper  electrode  is  positive  against  a  molar  solution 
of  copper  sulphate.  From  the  above  results,  it  is  possible  to  cal- 
culate the  electromotive  force  of  the  combination, 

Zn  -  m  ZnS04  1|  m  CuS04  -  Cu. 


ELECTROMOTIVE  FORCE  475 

Since,  according  to  Nernst's  equation, 

RT        Pl      RT,      P2 

E=El-E2=]og6--—loe-) 


where  PI  and  p\  refer  to  the  copper,  and  P2  and  pz  refer  to  the  zinc, 
we  have 

E  =  0.585  -  (  -  0.52)  =  1.105  volts. 

Concentration  Elements.  We  now  proceed  to  consider  cells  in 
which  the  electromotive  force  depends  primarily  on  differences 
in  concentration,  —  the  so-called  "  concentration  elements." 

Concentration  elements  may  be  conveniently  divided  into  two 
classes  :  (a)  elements  in  which  the  electrodes  are  of  different  concen- 
trations, and  (b)  elements  in  which  the  solutions  are  of  different 
concentrations. 

(a)  Elements  in  which  the  Electrodes  are  of  Different  Concentra- 
tions. (Amalgams  and  Alloys.)  If  in  the  equation 

RT,      Pi      RT,      P2 

E   =  —7T  log*  ---  FT  loge  —  > 

nF     3  pi       nF      3  p2 

p1  =  pZ)  as  is  the  case  when  the  ionic  concentrations  of  the  two 
solutions  are  identical,  then  we  have 


where  PI  and  P2  are  the  respective  solution  pressures  of  the  metal 
dissolved  in  the  electrodes.  If  the  amalgams  are  dilute,  the 
osmotic  pressure  of  the  dissolved  metal  will  be  proportional  to  the 
solution  pressure  of  the  electrode,  and  since  osmotic  pressure 
is  proportional  to  concentration,  we  may  replace  PI  and  P2,  in  the 
above  formula,  by  the  proportional  terms,  Ci  and  c2,  the  respec- 
tive  concentrations  of  the  metal  in  the  two  electrodes.  Hence, 
we  have 

_-.      RT  ,      c\  /IT\ 

E  =  ^lo&--  (17) 

The  accuracy  of  this  equation  has  been  fully  established  by  the 
experiments  of  Meyer,*  and  Richards  and  Forbes,  f 

*  Zeit.  phys.  Chem.,  7,  477  (1891). 

t  Publication  of  the  Carnegie  Institution,  No.  56. 


476 


THEORETICAL  CHEMISTRY 


Meyer's  results  for  zinc  amalgams  in  solutions  of  zinc  sulphate 
are  given  in  the  accompanying  table. 

E.  M.   F.   OF  ZINC  AMALGAMS 


T, 
degrees. 

Cl 

C2 

E  (obs.). 

E  (calc.). 

284.6 

0.003366 

0.00011305 

0.0419 

0.0416 

291.0 

0.003366 

0.00011305 

0.0433 

0.0425 

285.4 

0.002280 

0.0000608 

0.0474      . 

0.0445 

333.0 

0.002280 

0.0000608 

0.0520 

0.0519 

The  agreement  between  the  observed  and  calculated  values  of 
E  is  all  that  can  be  desired.  That  the  above  formula  holds  for 
zinc  amalgams  may  be  considered  as  a  proof  of  the  fact,  that  the 
zinc  dissolves  in  the  mercury  as  monatomic  molecules.  Thus, 
suppose  the  zinc  to  be  present  in  the  mercury  in  the  form  of  dia- 
tomic molecules;  then,  while  the  electrical  energy  would  be  equal 
to  2  FE,  the  osmotic  work  required  to  develop  this  energy  would 

be  -  R  T  loge  —  >  hence  we  should  have 
2  Cz 


(18) 


or,  the  calculated  value  of  the  electromotive  force  would  be  just 
one-half  of  the  observed  value.  The  mercury  in  the  amalgam 
has  been  shown  to  exert  no  effect  upon  the  electromotive  force  of 
the  cell,  so  long  as  the  dissolved  metal  has  the  greater  potential, 
(b)  Elements  in  which  the  Solutions  are  of  Different  Concentra- 
tions. In  this  type  of  cell  we  have  two  electrodes  of  the  same 
metal  immersed  in  solutions  of  different  ionic  concentrations  of 
the  metal.  Hence,  we  may  put  PI  =  P2  in  the  equation, 

RT ,       Pi      RT .      P2 


which  then  takes  the  form, 


pi 


Since  osmotic  pressure  is  proportional  to  concentration,  pi  and 
&  may  be  replaced  by  the  proportional  terms,  d  and  c?,  and  the 


ELECTROMOTIVE  FORCE  477 

foregoing  equation  becomes, 


or 

,_      RT 


where  mi  and  mz  are  the  molar  concentrations  of  the  two  solutions, 
and  ai  and  0:2  are  the  corresponding  degrees  of  ionization.  As  an 
example  of  a  concentration  element  of  this  class  we  may  take  the 
following  : 

Ag  -  0.01  m  AgNO3  ||  0.1  m  AgNO3  -  Ag. 

The  degrees  of  ionization  of  the  two  solutions  at  18°  are  as  follows  : 
-for  0.01  m  AgNO3,  a  =  0.93,  and  for  0.1  m  AgN02,  a  =  0.81. 
Substituting  in  the  equation, 

E  =  0.058  log  ^, 
m2<2 

we  have 


The  value  of  E,  found  by  direct  experiment,  is  0.055  volt. 

In  the  example  just  given,  the  electrodes  are  reversible  with 
respect  to  the  positive  ion  of  the  electrolyte.  Such  electrodes  are 
known  as  electrodes  of  the  first  type.  It  is  also  possible  to  construct 
cells  with  electrodes  which  are  reversible  with  respect  to  the  nega- 
tive ion  of  the  electrolyte.  These  are  termed  electrodes  of  the 
second  type.  The  calomel  electrode  is  an  example  of  an  electrode 
of  this  latter  type.  If  positive  electricity  passes  from  the  metal  to 
the  solution,  the  mercury  combines  with  the  Cl'  ions,  forming 
mercurous  chloride,  and  if  positive  electricity  passes  in  the  reverse 
direction,  chlorine  dissolves,  and  mercurous  chloride  is  formed. 
In  other  words,  the  electrode  behaves  like  a  chlorine  electrode, 
giving  up,  or  absorbing  the  element,  according  to  the  direction  of 
the  current.  A  typical  combination  involving  an  electrode  of 
the  second  type  is  the  following: 

Ag  -  0.1  m  AgN03  -  KNO3  -  0.1  m  KC1,  AgCl  -  Ag. 

This  particular  combination  was  studied  by  Goodwin,*  with  a 
*  Zeit.  phys.  Chem.,  13,  577  (1894). 


478  THEORETICAL  CHEMISTRY 

view  to  determining  the  solubility  of  silver  chloride.  If  we  assume 
a  saturated  solution  of  silver  chloride  to  be  completely  ionized 
then  the  solubility  product  will  be 

•i 
CAS'  X  Cci'  =  S. 

Since  the  concentrations  of  the  two  ions,  Ag*  and  Cl',  are  equal, 
it  follows  that  Vs  will  be  equal  to  the  solubility  of  the  silver 
chloride.  The  electromotive  force  of  a  concentration  cell  at  25° 
is  given  by  the  equation, 

00595 


U 

or 


En 


00595 

The  value  of  E,  at  25°,  for  the  above  cell  was  found  to  be  0.450  volt. 
The  degrees  of  ionization  of  the  two  electrolytes  are  as  follows  :  — 
for  0.1  molar  AgNO3,  a  =  0.82,  and  0.1  molar  KC1,  a  =  0.85. 
Substituting  in  the  preceding  equation,  we  obtain 

0.1  X  0.82  0.450 

C2  0.0595' 
therefore, 

cs  =  2.24  X  10-9. 

Or,  2.24  X  10~9  is  the  concentration  of  the  Ag*  ion  in  mols  per 
liter  in  a  0.1  molar  potassium  chloride  solution  of  silver  chloride. 
Hence,  the  solubility  product  s  will  be 

a  =  2.24  X  10-9  X  0.085  =  1.91  X  10~10, 
and 

Vs  =  1.38  X  10-5; 

that  is,  the  solubility  of  silver  chloride  in  a  saturated  aqueous 
solution  is  1.38  X  10~5  mol  per  liter  at  25°. 

The  Difference  of  Potential  at  the  Junction  of  the  Solutions 
of  Two  Electrolytes.  Thus  far  we  have  not  taken  into  consider- 
ation the  potential  differences  which  may  be  established  at  the 
junction  of  two  solutions.  Nernst*  has  shown,  that  in  many  cases 
it  is  possible  to  calculate  these  differences  of  potential  by  means 
of  his  osmotic  theory  of  the  origin  of  electromotive  force,  and  the 
values  thus  obtained  are  in  close  agreement  with  the  results  of  ex- 

*  Zeit.  phys.  Chem.,  4,  129  (1889). 


ELECTROMOTIVE  FORCE  479 

periment.  Let  us  imagine  that  two  solutions  of  hydrochloric  acid 
of  different  concentrations  are  brought  together  so  as  to  avoid  mix- 
ing, the  acid  in  each  solution  being  highly  ionized.  The  hydro- 
gen and  chlorine  ions  will  diffuse  independently,  and  since  the 
former  move  with  the  greater  velocity,  the  more  dilute  solution 
will  soon  contain  an  excess  of  H*  ions  and  the  more  concentrated 
solution  an  excess  of  Cl'  ions.  The  more  dilute  solution  will  be- 
come positively  charged,  owing  to  the  presence  of  an  excess  of  H" 
ions,  while  the  more  concentrated  solution  will  acquire  a  negative 
charge,  due  to  the  presence  of  an  excess  of  Cl'  ions.  The  accumu- 
lation of  positive  electricity,  however,  will  retard  the  velocity  of 
the  H*  ions  and  accelerate  the  velocity  of  the  Cl'  ions,  so  that 
ultimately,  the  two  ions  will  move  with  the  same  velocity.  The 
difference  of  potential  produced  in  this  way  will  cease  to  exist 
when  the  two  solutions  have  acquired  the  same  concentra 
tion. 

In  general,  it  may  be  stated,  that  the  difference  of  potential  set 
up  at  the  junction  of  the  two  solutions  is  caused  by  the  differ- 
ence in  the  rates  of  migration  of  the  two  ions,  the  more  dilute 
solution  acquiring  a  charge  corresponding  to  that  of  the  faster 
moving  ion. 

Electromotive  Force  of  Concentration  Cells  with  Transference. 
Let  u  and  v  be  the  migration  velocities  of  the  cation  and  the 
anion  respectively,  and  let  p\  be  the  osmotic  pressure  of  the  ions 
in  the  concentrated  solution,  and  p2  the  osmotic  pressure  of  the 
ions  in  the  dilute  solution.  Now  let  one  faraday  of  electricity  pass 

from  the  concentrated  to  the  dilute  solution;  then  —  ;  —   gram- 

u  +  v 

equivalents  of  positive  ions  will  migrate  from  the  concentrated 
to  the  dilute  solution,  while  •  gram-equivalents  of  negative 

ions  will  migrate  from  the  dilute  to  the  concentrated  solution. 
The  maximum  work  done  by  the  process  will  be 


e 

U  +  V  6  £2        U  +  V 

The  corresponding  electrical  energy  developed  is  EF,  hence  we 
have 


\ 


480  THEORETICAL  CHEMISTRY 

or,  since  osmotic  pressure  is  proportional  to  concentration,  we 
may  substitute  Ci  and  c%  for  pi  and  p2,  in  the  preceding  equation, 
and  obtain  the  following  expression  for  the  electromotive  force 
at  the  junction  of  the  two  solutions:  — 


As  an  example  of  the  use  of  the  above  formula,  we  may  take  the 
calculation  of  the  electromotive  force  of  the  following  combina- 
tion :  — 

Ag  -  0.1  m  AgNO3  -  0.01  m  AgN03  -  Ag. 

(a)  (b)  (c) 

Taking  the  junctions  (a),  (b),  and  (c)  in  order,  we  obtain 
RT  .       C-u-vRT.       ci      RT  .       C  * 

E  =  - 


2v      RT 


u  +  v    F 

=  2  na,  t 
therefore,  we  may  write  the  preceding  equation  in  the  form, 


But  —  :  —  =  2  na,  the  transference  number  of  the  anion,  and 
u  +  v 


.  (22) 

Assuming  the  temperature  to  be  17°,  and  passing  to  Briggsian 
logarithms,  we  have 

E  =  O.llGwalog--  (23) 

Cl 

The  transference  number  for  the  anion  of  silver  nitrate  is  0.522, 
while  ci  =  raiai  =  0.1  X  0.82  =  0.082,  and  eg  =  ra^  =  0.01  X 
0.94  =  0.0094.  Hence, 

0 
E  =  -  2  X  0.522  X  0.058  X  l°g 

or 

E  =  -  0.057  volt. 

The  value  found  by  direct  experiment  is  —  0.055  volt. 

It  is  to  be  noted,  that  if  the  electromotive  force  at  the  junction 
of  the  two  solutions  is  negative,  equation  (22)  takes  the  form, 

(24) 
i 

where  nc  is  the  transference  number  of  the  cation. 

*  C  is  a  number  proportional  to  the  solution  pressure  P, 


ELECTROMOTIVE  FORCE  481 

Electromotive  Force  of  Concentration  Cells  without  Trans- 
ference. A  less  familiar  type  of  concentration  cell  is  that  which 
does  not  involve  transference.  The  following  combination  may 
be  taken  as  an  example  of  such  a  cell: 

Ag  |  AgCl,  KC1  1  K  (Hg)4  1|  K  (Hg)4  1  KC1,  AgCl  |  Ag. 
Ci  Cz 

It  will  be  observed,  that  this  cell  is  in  reality  made  up  of  two 
independent  cells,  and  involves  no  liquid  junction.  While  the 
elimination  of  diffusion  appreciably  simplifies  the  theoretical 
treatment  of  cells  of  this  type,  practically,  it  is  difficult  to  find 
electrodes  which  are  reversible  toward  both  ions  of  the  electrolyte. 
The  passage  of  one  faraday  of  electricity  through  the  cell  results 
in  the  formation,  on  the  dilute  side,  of  one  equivalent  of  potassium 
chloride,  from  the  silver  chloride  and  the  amalgam,  while  on  the 
concentrated  side,  a  corresponding  amount  of  potassium  chloride 
is  decomposed.  If  we  assume  the  solutions  to  be  so  dilute  as  to 
be  completely  ionized,  one  equivalent  of  potassium  chloride  will 
function  as  two  mols,  and  the  electromotive  force  of  the  cell 
will  be 

.  (25) 


Formulas  for  the  Difference  of  Potential  at  Liquid  Junctions. 

As  has  been  pointed  out,  the  total  electromotive  force  of  a  con- 
centration cell  with  transference  is  made  up  of  the  algebraic  sum 
of  the  potentials  at  the  two  electrodes,  and  the  potential  at  the 
junction  of  the  two  solutions.  Since  the  value  of  the  potential  at 
the  electrodes  alone  is  often  desired,  numerous  formulas  have  been 
derived  for  the  calculation  of  the  potential  at  the  liquid  junction. 

In  addition  to  the  formula  of  Nernst,  already  mentioned,  for- 
mulas have  been  proposed  by  Planck,*  Henderson,!  Gumming,  J 
Lewis  and  Sargent,  §  and  Maclnnes.||  Of  these  different  formulas, 
that  of  Maclnnes  possesses  distinct  advantages. 

If  one  faraday  of  electricity  be  passed  through  the  cell, 
Ag  |  AgCl,  KC1  1  KC1,  AgCl  |  Ag, 

Cl  Cz 

*  Wied.  Ann.,  40,  561  (1890). 

f  Zeit.  phys.  Chem.,  59,  118  (1906);  63,  325  (1908). 

t  Trans.  Faraday  Soc.,  8,  86  (1912);  9,  174  (1913). 

§  Jour.  Am.  Chem.  Soc.,  31,  363  (1909). 

||  Ibid.,  37,  2301  (1915). 


482  THEORETICAL  CHEMISTRY 

one  equivalent  of  chloride  ions  will  enter  the  dilute  solution,  while 
a  corresponding  amount  will  be  electrolyzed  out  of  the  more  con- 
centrated solution.  The  current  will  be  carried  across  the  liquid 
junction  by  the  migration  of  nc  equivalents  of  potassium  ions  in 
the  direction  of  the  current,  and  by  the  migration  of  (1  —  nc) 
equivalents  of  chloride  ions  in  the  opposite  direction.  The  total 
effect  of  the  passage  of  one  faraday  of  electricity,  will  be  the  trans- 
ference of  nc  equivalents  of  salt  from  the  more  concentrated  to  the 
more  dilute  solution.  The  osmotic  work  at  the  junction  of  the 
two  solutions  will  correspond  to  the  algebraic  sum  of  the  number 
of  ion  equivalents  which  have  migrated  from  the  concentrated  to 
the  dilute  solution,  or  nc  —  (1  —  nc)  =  2  nc  —  1. 

In  order  to  obtain  the  electrical  energy  necessary  to  perform 
this  amount  of  osmotic  work,  let  us  consider  the  following  cell 
involving  no  transference  : 

Ag  |  AgCl,  KC1    K  (Hg),  ||  K  (Hg)4  j  KC1,  AgCl  |  Ag. 

Ci  C2 

The  passage  of  one  faraday  of  electricity  through  this  cell  in- 
volves, as  we  have  seen,  the  formation  of  one  equivalent  each  of 
potassium  and  chloride  ions  in  the  dilute  solution,  and  the  re- 
moval of  one  equivalent  of  each  ion  from  the  more  concentrated 
solution.  The  electrical  energy  accompanying  the  transfer  of  two 
ion  equivalents,  from  one  solution  to  the  other,  will  be  equal  to 
the  product  of  the  electromotive  force  of  the  cell  and  1  faraday, 
or  EF. 

The  electromotive  force  EL,  at  the  liquid  junction,  can  now  be 
obtained  by  the  simple  proportion, 

EF  :  ELF  =  2  :  2  nc  -  1, 
*-*<**-».  (26) 

Since  the  ratio  of  Et,  the  electromotive  force  of  the  cell  with  trans- 
ference, to  E,  the  electromotive  force  of  the  cell  without  trans- 
ference, is  equal  to  ne,  it  follows,  that  equation  (26)  may  be  written 
in  the  form,  • 


It  is  to  be  observed,  that  equations  (26)  and  (27)  involve  no  assump- 
tions concerning  the  concentrations  of  the  ions  in  the  solutions,  one 


ELECTROMOTIVE  FORCE  483 

of  the  characteristics  which  distinguishes  these  formulas  from  any 
of  the  others  hitherto  proposed. 

Lewis  and  Sargent  *  have  proposed  a  modification  of  a  formula 
derived  by  Planck,  for  the  difference  of  potential  at  the  junction  of 
two  equally  concentrated  electrolytes  having  a  common  ion.  If 
AI  and  A2  are  the  equivalent  conductances  of  the  two  electrolytes, 
the  value  of  the  liquid^  junction  potential,  EL)  is  given  by  the 
formula, 

&-^l°*£-  (28) 

The  junction  between  two  different  concentrations  of  two  elec- 
trolytes with  a  common  ion  may  be  replaced  by  two  junctions, 
one  of  which  connects  two  different  concentrations  of  the  same 
electrolyte,  and  the  other  connects  two  different  electrolytes  of  the 
same  concentration.  For  example,  the  junction, 

0.1NNaCl|0.05NHCl 
may  be  replaced  by 

0.1  N  NaCl  |  0.05  N  Nad  |  0.05  N  HC1, 

(A)  (B) 

in  which  the  potential  of  the  junction  (A)  may  be  calculated  by 
equations  (26)  or  (27) ,  and  the  potential  of  junction  (B)  by  equa- 
tion (28). 

The  electromotive  force  at  the  junction  of  solutions  of  differently 
concentrated  uni-univalent  ionogens  which  do  not  contain  a  com- 
mon ion  may,  in  like  manner,  be  calculated  by  employing  three 
junctions.  Thus,  the  junction, 

0.1NKNO3|0.05NNaCl, 
may  be  replaced  by 

0.05  N  NaCl    0.1  N  NaCl  |  0.1  N  KC1 1  0.1  N  KNO3. 
(A)  (B)  (BO 

The  potential  at  (A)  may  be  calculated  by  means  of  equations  (26) 
or  (27),  and  the  potentials  at  (B)  and  (B')  by  means  of  equation 
(28). 

By  the  use  of  equations  (26),  (27),  and  (28),  in  the  manner  indi- 
cated, it  is  possible  to  calculate  the  junction-potential  between  the 
solutions  of  any  two  uni-univalent  ionogens,  to  within  a  few  tenths 
of  a  millivolt. 

*  loc.  cit. 


484 


THEORETICAL  CHEMISTRY 


Flowing  Junctions.  The  importance  of  renewing  the  surface 
at  the  junction  of  the  solution  of  two  electrolytes,  in  order  to  secure 
constant  and  reproducible  differences  of  potential,  has  been  em- 
phasized by  a  number  of  investigators,  notably  by  Walpole,* 
who  pointed  out  that  fairly  constant  values  could  be  obtained  by 

forming  the  junctions  with  tapes 
along  which  the  solutions  were 
permitted  to  flow. 

Lamb  and  Larson  f  were  the  first, 
however,  to  develop  a  "  flowing 
junction,"  by  means  of  which  dif- 
ferences of  potential,  constant  and 
reproducible  to  within  0.01  milli- 
volt, can  be  obtained.  This  result 
was  secured  by  allowing  an  upward 
current  of  the  denser  electrolyte  to 
meet  a  downward  current  of  the 
lighter  electrolyte,  in  a  vertical 
tube  at  its  point  of  union  with 
a  horizontal  outflow  tube,  as  is 
shown  diagrammatically  in  Fig.  122. 
Recently  Maclnnes  and  Yeh,  t 
employing  a  modified  form  of  the 
Lamb  and  Larson  flowing  junction, 
have  shown  that  the  values  of  the 


Fig.  122 


junction  potential,  calculated  by  the  formula  of  Lewis  and  Sargent, 
(equation  (28)),  are,  in  nearly  every  case,  in  excellent  agreement 
with  those  found  by  direct  experiment. 

Normal  Electrode  Potential.  The  difference  of  potential  at 
a  reversible  electrode,  when  the  concentration  of  the  ions  of  the 
electrolyte  is  normal,  is  known  as  the  normal  electrode  potential. 

According  to  Nernst,  the  expression  for  the  difference  of  poten- 
tial at  a  single  electrode  is 

P 


(29) 


RT,      C 


*  Jour.  Chem.  Soc.,  105,  2521  (1914). 
f  Jour.  Am.  Chem.  Soc.,  42,  229  (1920) 
j  Ibid.  43,  2563  (1921). 


ELECTROMOTIVE  FORCE 


485 


where  P  is  the  solution  pressure  of  the  electrode,  p  is  the  osmotic 
pressure  of  the  ions  of  the  electrolyte,  and  where  C  and  c  are  con- 
centrations proportional  to  P  and  p  respectively.  If  the  concen- 
tration of  the  ions  is  unity,  c  =  1,  and  the  expression  for  the  normal 
electrode  potential  E0  becomes 

r>/77 

Et>=w\0geC.  (30) 

On  subtracting  (29)  from  (30),  we  obtain, 

E0  =  E  H — ^-logec,  (31) 

where  c  is  the  concentration  of  the  Ions  in  a  solution  whose  elec- 
trode potential  is  E.  In  the  actual  determination  of  normal  elec- 
trode potentials,  the  value  of  c,  in  equation  (31),  is  usually  very 
much  less  than  unity. 

The  following  table  gives  the  values  of  most  important  elec- 
trode potentials,  the  data  in  the  second  column  being  referred  to 
the  normal  calomel  electrode  as  zero,  while  the  data  of  the  third 
column  is  referred  to  the  normal  hydrogen  electrode  as  zero. 

NORMAL  ELECTRODE  POTENTIALS  AT  25° 


Electrode 

Normal  Calomel 
Electrode  =  0 

Normal  Hydrogen 
Electrode  =  0 

Authority 

Li,  Li'  

3.3044 

i/    3.0216 

Lewis  &  Kraus  (1) 

Na,  Na'  

2.9981 

2.7153 

Lewis  &  Kraus  (2) 

K,  K'  

3.2084 

2.9256 

Lewis  &  Keyes  (3) 

Mg,  Mg".  .  . 

1.83 

1.55 

Zn,  Zn".... 

1.04 

0.76 

Fe,  Fe".... 

0.71 

0.43 

Cd,  Cd".... 

0.68 

0.40 

Tl,  Tl*  

0.617 

0.334 

Lewis  &  von  Ende  (4) 

Co,  Co".... 

0.573 

0.29 

Ni,  Ni".... 

0.50 

0.22 

Sn,  Sn"  

0.4263 

0.1435 

Pb,  Pb".... 

0.4125 

0.1295 

Getman  (5) 

H2rH'  

0.2828 

0.0000 

~Cu,  Cu".... 

-0.060 

-0.340 

Lewis  &  Lacey  (6) 

Cu,  Cu'  .  .  .  . 

-0.230 

-0.510 

Ag,  Ag'  
Hg,  Hg".... 

-0.516 
-1.14 

-0.799 
-0.86 

Noyes  &  Brann  (7) 

I,  I'.    

-0.250 

-0.533 

Br2,  Br'  

-0.8039 

-1.0872 

Lewis  &  Storch  (8) 

C12,  Cl'.. 

-1.0795 

-1.3623 

Lewis  &  Rupert  (9) 

02,  OH'.... 

-0.110 

-0.393 

Lewis  &  Kraus  (2) 

(1)  Jour.  Am.  Chem.  Soc.,  35,  341  (1913).  (2)  Ibid.  32,  1459  (1910). 
(3)  Ibid.  34,  119  (1912).  (4)  Ibid.  32,  732  (1910).  (5)  Ibid.  40,  619  (1918). 
(6)  Ibid.  36,  804  (1913).  (7)  Ibid.  34, 1016  (1912).  (8)  Ibid.  39, 1910  (1917). 
(9)  Ibid.  33,  299  (1911). 


486  THEORETICAL  CHEMISTRY 

Electrometric  Determination  of  Valence.  The  equation  of 
Nernst,  for  the  electromotive  force  of  a  concentration  cell,  may  be 
employed  to  determine  the  valence  of  the  ions. 

For  example,  the  chemical  behavior  of  mercurous  salts  is  such 
as  to  justify  the  use  of  single  or  double  formulas,  involving  either 
Hg*  or  Hg2".  To  determine  which  of  these  two  formulas  is  correct, 
Ogg  *  set  up  the  following  cell  : 

Hg  |  0.5  N  Mercurous  Nitrate  ||  0.05  N  Mercurous  Nitrate  |  Hg 
0.1NHNO3  0.1NHNO3 

and  found  its  electromotive  force,  at  17°,  to  be  0.029  volt.  If  we 
neglect  the  difference  of  potential  at  the  junction  of  the  two  solu- 
tions, the  electromotive  force  of  this  cell  may  be  calculated  by  the 
familiar  formula, 

T-T      RT,      C2 

E  =  —pj  loge  ~  • 

nF     &  ci 

Assuming  that  GZ/CI  =  10,  and  passing  to  Briggsian  logarithms, 
we  have 

0.029  =  0.058/rc, 
or  n  =  2. 

Hence,  the  valence  of  the  mercurous  ion  is  2  and  it  must,  in  con- 
sequence, be  represented  by  Hg2".  It  follows,  that  the  correct  for- 
mula of  mercurous  nitrate  is  Hg2(NO3)2. 

Electrometric  Determination  of  Transference  Numbers.  The 
formula  for  the  electromotive  force  of  a  concentration  cell  with 
transference  has  been  shown  to  be 


If  the  electromotive  force  of  such  a  combination  is  measured,  and 
the  values  of  Ci  and  c%  are  accurately  known,  obviously  the  value 
of  the  transference  number  of  the  cation,  ne,  can  be  calculated.  If, 
after  having  measured  Et,  a  similar  cell  without  transference  be 
set  up,  and  its  electromotive  force  measured,  it  is  possible  to  obtain 
an  expression  for  nc  which  does  not  involve  either  Ci  or  c2.  As  has 
been  shown,  the  electromotive  force  of  a  cell  without  transference, 
E,  is  given  by  the  formula, 


*  Zeit.  phys.  Chem.,  27,  285  (1898). 


ELECTROMOTIVE  FORCE 


487 


Dividing  equation  (32)  by  equation  (33),  we  obtain 

E, 

*•*"%' 


(34) 


which  expression  gives  the  transference  number  in  terms  of  the  two 
measured  electromotive  forces.  This  method  for  the  determina- 
tion of  transference  numbers  was  first  suggested  by  Helmholtz.* 
The  transference  number  of  the  lithium  ion  has  recently  been  de- 
termined in  this  manner  by  Pearce  and  Mortimer,  f  using  cells 
made  up  according  to  the  following  schemes: 

Ag  |  AgCl,  LiCl  ||  LiCl,  AgCl  |  Ag  (with  transference), 

Ci  C2 

Ag  |  AgCl,  LiCl  Li  (Hg)*  ||  Li  (Hg)«  |  LiCl,  AgCl  |  Ag  (without 

d  C2  transference). 

A  comparison  between  their  results  and  the  values  given  by 
Kohlrausch  and  Holborn  |  is  afforded  by  the  accompanying  table. 

Concentration  Ratio 1.0-0.1    0.5-0.05    0.1-0.01     0.05-0.005 

Mean  nc  (K.  and  H.) 0.285          0.300          0.340  0.360 

Meannc  (P.  and  M.) 0.279          0.322          0.343  0.365 

It  will  be  observed  that  the  agreement  between  the  two  series^of 
results  is  satisfactory. 

Later  determinations  of  the  transference  numbers  of  lithium 
chloride  solutions,  by  Maclnnes  and  Beattie,  §  are  given  in  the 
accompanying  table. 

TRANSFERENCE  NUMBERS  OF  LITHIUM  ION 
IN  LITHIUM  CHLORIDE 


Concentration 

E.  m.  f.  Method 

"  Best  value  " 
Hittorf  Method 

0.001 

0.359 



0.005 

0.341 

. 

0.01 

0.334 

0.332 

0.02 

0.327 

0.328 

0.05 

0.318 

0.320 

0.10 

0.311 

0.313 

0.20 

0.304 

0.304 

0.30 

0.299 

0.299 

0.50 

0.293 



1.0 

0.286 



2.0 

0.276 



3.0 

0.268 

*  Ges.  Abhl.  I,  840:  II,  979.  t  Leitvermogen  der  Elektrolyte,  p.  201. 

t  Jour.  Am.  Chem.  Soc.,  40, 518  (1918).  §  Jour.  Am.  Chem.  Soc.,  42, 1117  (1920). 


488  THEORETICAL  CHEMISTRY 

These  results  have  been  shown  to  be  in  close  agreement  with  the 
direct  transference  measurements  of  Jahn,  Bein,  and  Washburn. 
Electrometric  Determination  of  Hydrolysis.  One  of  the  most 
satisfactory  methods  which  we  possess  for  the  determination  of  the 
degree  of  hydrolysis  of  salts  depends  upon  the  measurement  of  the 
electromotive  force  of  cells  made  up  as  follows : 

PtH2 1  salt  solution  |  sat.  NH4NO3*  ||  N  KC1,  Hg2Cl2 1  Hg. 

i 

From  the  measured  electromotive  force  of  the  cell,  the  concentra- 
tion of  the  hydrogen  ion  in  the  solution  is  determined,  and 
from  this  the  degree  of  hydrolytic  dissociation  of  the  salt  can  be 
calculated.  The  method  is  especially  valuable  in  cases  where  the 
concentration  of  the  hydrogen  ion  is  very  small.  Unfortunately, 
the  application  of  the  method  is  restricted  to  the  salts  of  metals  less 
noble  than  hydrogen.  In  other  words,  it  cannot  be  used  to  deter- 
mine the  hydrolysis  of  the  salts  of  metals  which  would  be  precipi- 
tated upon  the  platinum  electrode.  The  applicability  of  the 
method  is  further  limited  by  the  fact,  that  certain  ions,  such  as  Fe"", 
N03',  and  CKV,  are  either  wholly,  or  partially  reduced  by  the 
hydrogen  of  the  hydrogen  electrode. 

The  method  has  been  successfully  applied,  by  Denham,  f  to 
the  determination  of  the  hydrolysis  of  various  salts,  among  which 
may  be  mentioned,  aluminium  chloride,  aluminium  sulphate, 
nickel  chloride,  nickel  sulphate,  cobalt  sulphate,  and  ammonium 
chloride.  As  an  illustration  of  the  method,  we  may  take  the  case 
of  ammonium  chloride  which  dissociates  hydrolytically,  according 
to  the  equation, 

NH4C1  +  H-OH  +±  NH4OH  +  HC1. 

Here  we  have  a  weak  base  and  a  very  strong  acid,  as  the  products 
of  hydrolytic  dissociation.  While  the  ammonium  hydroxide  may 
be  regarded  as  practically  non-ionized,  the  hydrochloric  acid  is  to 
be  considered  as  having  undergone  complete  ionization.  If  one 
mol  of  ammonium  chloride  is  dissolved  in  v  liters  of  water,  and  the 
degree  of  hydrolytic  dissociation  is  x,  then  the  concentrations  of 
the  products  of  the  reaction,  ammonium  hydroxide  and  hydro- 

*  A  saturated  solution  of  ammonium  nitrate  is  interposed  between  the 
solutions  to  eliminate  any  difference  of  potential  at  the  junction  of  the  solu- 
tions. 

t  Jour.  Chem.  Soc.,  93,  41  (1908);  Zeit.  anorg.  Chem.,  57,  361  (1908). 


ELECTROMOTIVE  FORCE  489 

chloric  acid,  will  be  x/v.  Since  the  acid  is  completely  ionized,  x/v 
will  also  represent  the  concentrations  of  the  hydrogen  and  chloride 
ions.  Assuming  that  the  active  mass  of  the  water  remains  con- 
stant, we  have,  according  to  the  law  of  mass  action, 


where  Kh  is  the  hydrolytic  constant.  Since  v  is  known,  it  only  re- 
mains to  determine  x/v,  or  the  concentration  of  the  hydrogen  ions, 
in  order  to  be  able  to  calculate  Kh.  The  potential  at  the  hydrogen 
electrode  is  given  by  the  familiar  formula, 


7P77 
or,  denoting  the  normal  electrode  potential,  --  =•  loge  C,  by  EQ, 

Ur 

we  may  write 

P?T  T 

.  (36) 


According  to  Denham,  the  electromotive  force  of  the  cell, 


PtH2 1  N/32  NH4C1 1  sat.  NH4NO3 1|  N  KC1,  Hg2Cl2 1  Hg, 

at  25°,  is  0.6056  volt,  the  current  flowing  outside  the  cell  in  the 
direction  of  the  arrow.  The  potential  of  the  normal  calomel 
electrode  being  +  0.56  volt,  it  follows,  that  the  potential  of  the 
hydrogen  electrode,  E  =  0.56  -  0.6056  =  -  0.0456  volt.  There- 
fore, 

-  0.0456  »  £•>+§£  loge  jf- 

The  absolute  value  of  the  potential  of  the  normal  hydrogen  elec- 
trode EQ,  referred  to  the  normal  calomel  electrode  as  +  0.56  volt, 
and  not  as  zero,  is  +  0.277  volt.  Substituting  this  value  in  the 
foregoing  equation,  and  passing  to  Briggsian  logarithms,  we  have 
0.059  log  x/v  =  -  0.0456  -  0.277  =  -  0.3226. 

Solving  this  equation,  we  find  x/v  =  0.3406  X  10"6  gram-ions  of 
hydrogen  per  liter.  Since  the  value  of  x/v  would  be  ^,  if  the  salt 
were  completely  hydrolyzed,  the  percentage  of  hydrolysis  under 
the  conditions  of  the  experiment,  namely,  when  v  =  32,  is 

0.3406  X .10-*  X  100  =0.0109percent. 


490  THEORETICAL  CHEMISTRY 

The  value  of  the  hydrolytic  constant  is  given  by  the  equation, 

x-  (0.3406  X  IP"6)2         -0363X10-14 

P*  ~  (i-x)v  ~  32  (1  -  0.3406  X  1Q-6) 

The  degree  of  hydrolytic  dissociation  of  ammonium  chloride  has 
been  determined,  by  Noyes,  from  measurements  of  electrical  con- 
ductance. The  value  of  x,  for  0.01  N  NH4C1,  was  found  by  him  to 
be  0.02,  at  18°,  while  Denham  found,  x  =  0.018,  at  25°,  by  extra- 
polation of  his  electrometric  data. 

Chemical  Cells.  The  source  of  the  electromotive  force  in  con- 
centration cells  is  the  diffusion  of  the  electrolyte,  from  a  region  of 
higher,  to  a  region  of  lower  concentration.  In  this  process,  there 
is  nothing  which  can  be  characterized  as  a  chemical  change.  On 
the  other  hand,  there  are  numerous  galvanic  combinations  in 
which  the  electromotive  force  is  due,  primarily,  to  chemical  reac- 
tions within  the  cells.  One  of  the  most  familiar  examples  of  this 
type  of  cell  is  the  Daniell  Cell,  which  may  be  represented  by  the 
following  scheme: 

Zn|ZnS04||CuS04iCu. 

The  reaction  taking  place  within  the  cell  may  be  written  ionically 
as  follows  : 

Zn  +  Cu"  -»  Zn"  +  Cu, 

and  the  equilibrium  constant  may  be  expressed  by  the  equation, 

T^         C  Zn"   X  C  cu  ff><7\ 

=  c'     .»  V  c'     '  ™7' 

C  Cu       X  C  zn 

where  the  terms  involving  c'  denote  equilibrium  concentrations. 
Since  the  concentrations  of  the  solid  metals  are  obviously  constant, 
equation  (37)  may  be  simplified  as  follows  : 

(38) 


Cu" 


If  J£0zn  and  EQCU  denote  the  normal  electrode  potentials  of  zinc 
and  copper,  respectively,  then  according  to  equation  (31)  we  will 
have 


(39) 

and 

(40) 


ELECTROMOTIVE  FORCE  491 

Therefore,  the  total  electromotive  force  of  the  cell  will  be 


E  =  Eza  -  Ec»  =  EOZD  -  £0-        log          •        (41) 

When  the  four  substances  Zn,  Cu,  Zn"  and  Cu"  are  in  equilibrium, 
equation  (41)  may  be  written  in  the  form, 

:  (42) 

According  to  van't  Hoff,  the  maximum  work  which  can  be 
derived  from  a  given  chemical  reaction  at  constant  temperature  is 
given  by  the  expression, 

W  =  RTlo&  K  -  RTZn  loge  C,  (see  p.  315)  (43) 

where   2  n  =  n\   +  n^'  +  •  •  •  —  n\  —  n^  —  -  •  •  in  the  general 
reaction  equation, 

-f-  nzAz  -f-  •  •  •  =  ni'Ai  +  n^A*  +  •  •  •  , 


and  where  C  denotes  all  the  arbitrary  concentration  terms. 

If  the  concentrations  of  the  zinc  and  copper  ions  are  Czn"  and 
ecu",  respectively,  then  the  maximum  work  obtainable  from  the 
reaction  taking  place  within  the  Daniell  cell  will  be,  according 
to  equation  (43), 

W  =  2  EF  =  RTloge^-  -  RTloge  —  •  (44) 

C  Cu"  Ccu" 

If  the  concentrations  of  the  solutions  of  the  sulphates  of  zinc  and 
copper  be  so  chosen  that  the  concentrations  of  the  ions  are  equal, 
then  the  second  term  of  equation  (44)  vanishes,  and  we  have 

W  =  2EF  =  RTloge  K.  (45) 

Since  at  25°,  E  =  1.1  volt  approximately,  we  may  substitute  these 
values  in  equation  (45),  and  evaluate  the  equilibrium  constant,  K, 
as  follows: 

1          V         2  X    L1         Q7 

log  K  -    -0059-   =  37> 
or  K=  1037. 

That  is,  when  equilibrium  is  established,  the  concentration  of  Zn" 
is  1037  times  that  of  Cu",  or  in  other  words,  the  solution  pressure 
of  zinc  is  1037  times  that  of  copper. 

Another  type  of  chemical  cell,  is  that  in  which  the  electrodes 


492  THEORETICAL  CHEMISTRY 

take  no  part  in  the  chemical  reactions  occurring  within  the  cell, 
but  merely  function  as  inert  electrical  conductors.  Let  the 
following  reaction  be  assumed  to  take  place  within  a  cell: 

2  FeCl3  +  SnCl2  <±  2  FeCl2  +  SnCl4. 
Expressed  in  terms  of  the  ions  involved,  this  becomes, 

2  Fe*"  +  Sn"  <=±  2  Fe"  +  Sn"". 
Let  the  cell  be  set  up  according  to  the  scheme, 


(Pt) 


Sn" 
Sn"" 


Fe" 
Fe'" 


(Pt), 


the  solution  of  ferrous  and  ferric  chlorides  forming  one-half  of  the 
cell,  while  the  solution  of  stannous  and  stannic  chlorides  forms  the 
other  half  of  the  cell.  When  the  platinum  electrodes  are  con- 
nected by  a  wire,  Fe'"  will  be  reduced  to  Fe",  charging  the  electrode 
positively,  while  on  the  other  side,  positive  electricity  will  enter 
by  the  electrode,  and  oxidize  Sn"  to  Sn"". 

According  to  equation  (42),  the  electromotive  force  of  the  cell 
will  be 

RT  RT       cFe2..  X  can—  ,AR. 

E  =  2F\ogeK  -  _  log.  £-5-^—  ,  (46) 

£/      2t<    \s    c' 

where  K  is  the  value  of    ,e2"  —  —  —  r~  ,  when  the  four  ions  Sn", 

C  Fe  •"    X  C  sn" 

Sn'"',  Fe"  and  Fe*"  are  in  equilibrium. 

Oxidation  and  Reduction  Elements.  When  a  dissolved  sub- 
stance passes  from  a  lower  to  a  higher  state  of  oxidation,  the 
change  in  the  positive  ion  may  be  considered  as  due  to  an  increase 
in  the  number  of  electrical  charges  on  the  ion.  Thus,  when  a  fer- 
rous salt  is  oxidized  to  the  ferric  state,  the  change  may  be  repre- 
sented by  the  equation, 


Similarly,  the  reduction  of  a  ferric  salt  to  the  corresponding  ferrous 
salt  may  be  represented  by  the  reverse  equation,  or 

Fe"'  -*  Fe"  +  (+). 

The  formation  of  zinc  ions  from  metallic  zinc  may  be  considered 
as  an  oxidation,  and  may  be  represented  by  the  equation, 

Zn  +  2  (+)  ->  Zn". 


ELECTROMOTIVE  FORCE  493 

The  formation  of  negative  ions  from  a  non-metallic  element  may 
be  considered  as  a  reduction,  as  for  example,  the  change  of  potas- 
sium ferricyanide  to  potassium  ferrocyanide,  which  may  be  rep- 
resented by  the  equation, 

Fe(CN)6'"  +  (-)  ->Fe(CNV"'. 

The  foregoing  considerations  lead  to  the  following  definition  of 
the  terms,  oxidation  and  reduction:  Oxidation  is  the  process  in 
which  a  substance  takes  up  positive,  or  parts  with  negative  charges, 
while  reduction  is  the  process  in  which  a  substance  takes  up  negative, 
or  parts  with  positive  charges. 

According  to  this  definition,  oxidation  always  takes  place  at 
the  anode  and  reduction  at  the  cathode. 

The  oxidizing  or  reducing  power  of  different  electrolytes  may 
readily  be  compared  by  measuring  the  resulting  electromotive 
force,  when  the  given  substances  are  made  the  active  agents  in  a 
cell.  For  example,  if  a  platinum  electrode  be  immersed  in  a  so- 
lution containing  both  a  ferrous  and  a  ferric  salt,  and  this  in  turn, 
be  connected  with  a  calomel  electrode  as  the  other  half-element, 
we  will  have  a  cell,  the  electromotive  force  of  which  will  furnish 
a  measure  of  the  oxidation,  or  reduction,  process  occurring  within  it. 
The  foregoing  cell  may  be  conveniently  represented  by  the  scheme, 

Hg2Cl2 


(Pt) 


Fe" 


Hg2" 


(Hg). 


If  we  assume  that  ferric  ions  are  reduced  to  ferrous  ions,  the 
direction  of  the  current  within  the  cell  will  be  from  right  to  left, 
and  the  process  may  be  represented  by  the  equation, 

2  Fe'"  +  Hg2  -»  Hg2*'  +  2  Fe". 
When  eauilibrium  is  reached,  the  equilibrium  constant  will  be 

(47) 


cW-  X  c'Hg2 
but  c'Hg2'«  and  c'Hg2  are  constant,  hence  we  have 

K'=^£-,  (48) 

C  Fe  ••• 

and  therefore, 

W  =  RT  lo&K'  -  RT\oge  ^  •  (49) 


494  THEORETICAL  CHEMISTRY 

Since  the  reaction  represented  by  the  above  equation  corresponds 
to  the  passage  of  two  faradays  of  electricity,  we  have,  since 
W  =  nEF, 


D/TT 

If  we  replace  75-™  log*  ^'  ^Y  E0,  the  preceding  equation  becomes, 


or  E-tf,-log.-  (51) 


The  value  of  the  electromotive  force  of  the  cell,  when  the  concentra- 
tions of  Fe'*  and  Fe'"  are  equal,  is  EQ}  the  latter  being  called  the 
normal  potential  of  the  process.  v 

If  instead  of  the  reduction  reaction,  we  had  considered  the 
oxidation  reaction, 

2  Fe"  +  Hg2"  ->  2  Fe'"  +  Hg2, 

we  would  have  derived  as  the  expression  for  the  oxidation  po- 
tential, 

(52) 


(Let  the  student  derive  equation  (52)). 

All  reversible  oxidation  and  reduction  processes  are  similarly 
related  to  each  other,  and  the  following  equation  gives  the  general 
relationship  between  ionic  concentrations  and  the  resulting  elec- 
tromotive force  : 


where  E(CQ  -»  a]  is  the  electromotive  force  produced  by  the  passage 
from  the  lower,  or  "  ous  "  state  of  oxidation,  to  the  higher,  or  "  ic  " 
state  of  oxidation;  EQ  is  the  normal  potential,  CQ  and  d  are  the 
ionic  concentrations  of  the  lower  and  higher  states  of  oxidation, 
respectively,  and  n  is  the  difference  in  valence  of  the  two  kinds  of 
ions.  A  number  of  oxidation  and  reduction  elements  have  been 
studied  by  Bancroft,*  Peters,  f  and  Kniipffer  and  Bredig.t 

lonization  and  Activity.     If  in  the  equation  for  the  electro- 
motive force  of  a  concentration  cell,  we  substitute  the  observed 

*  Zeit.  phys.  Chem.,  10,  387  (1892).  f  Ibid.  26,  193  (1895). 

J  Ibid.  26,  255  (1895). 


ELECTROMOTIVE  FORCE 


495 


value  of  E,  and  solve  the  equation  for  the  ratio,  <k/Ci,  we  find  that 
the  value  thus  obtained  does  not  agree  with  the  value  derived 
from  conductance  measurements.  In  other  words,  if  MI  and  mz 
are  the  actual  molar  concentrations  of  the  two  solutions  forming 
the  cell,  and  a\  and  «2  are  the  corresponding  degrees  of  ioniza- 
tion,  we  find  that  the  ratio,  m^c^/miai,  is  not  equal  to  the  ratio, 
C2/Ci,  as  calculated  from  the  electromotive  force  of  the  cell.  This 
is  shown  by  the  following  table,  which  gives  the  ratios  of  the 
ionic  concentrations  of  solutions  of  potassium  chloride  correspond- 
ing to  various  ten-to-one  salt  concentrations. 

RATIOS  OF  ION  CONCENTRATIONS  IN  SOLUTIONS  OF 
POTASSIUM  CHLORIDE 


Concentration 
Ratio. 

C2/Cl 

(Conductance). 

C2/Ci 

(Electromotive 
Force). 

0.5  :0.05 
0.1  :0.01 
0.05:0.005 
0.01:0.001 

8.85 
9.16 
9.30 
9.62 

8.09 
8.33 

8.64 
9.04 

The  figures  in  the  last  column  of  the  table  are  known  as  activity 
ratios,  and  the  individual  values  of  c\  and  fy  are  termed  activi- 
ties. If  it  be  assumed  that  the  activities  of  the  ions  at  the  low- 
est concentrations  at  which  accurate  measurements  can  be  made; 
viz.,  from  0.001  to  0.003  molal,  are  identical  with  the  corresponding 
conductance  ratios,  it  follows  that  the  activities  at  other  concen- 
trations can  be  calculated.  The  rajio  of  the  activity  to  the  con- 
centration is  known  as  the  "  thermodynamically  effective " 
ionization,  or  the  "  activity  coefficient." 

The  following  table,  compiled  by  Noyes  and  Maclnnes,*  con- 
tains the  activity  coefficients  of  four  typical  uni-univalent  electro- 
lytes, together  with  the  corresponding  values  of  the  conductance 
ratio. 

The  differences  between  the  two  sets  of  values  are  much  too 
large  to  be  attributed  to  experimental  error.  Furthermore,  it 
will  be  observed  that  there  is  a  distinct  minimum  in  the  values 
of  the  activity  coefficients  of  lithium  chloride,  hydrochloric  acid 
and  potassium  hydroxide,  at  approximately  0.5mola}  concentration. 
The  existence  of  this  minimum  is  clearly  shown  by  the  curves  in 
Fig.  123,  which  are  plotted  from  the  data  of  the  table,  and 
*  Jour.  Am.  Chem.  Soc.,  42,  239  (1920). 


496 


THEORETICAL  CHEMISTRY 


ACTIVITY  COEFFICIENTS  AND  CONDUCTANCE- 
VISCOSITY   RATIOS 


Mols  per 

Activity  Coefficients 

Conductance-Viscosity  Ratios 

1000  g.  of 
Water 
Cone. 

KC1 

LiCl 

HCl 

KOH 

KC1 

LiCl 

HCl 

KOH 

0.001 

0.979 

0.976 

0.979 

0.976 

0.990 

0.003 

0.943 

0.945 

0'.990 

6^982 

0.968 

0.962 

0.986 

6.980 

0.005 

0.923 

0.930 

0.965 

0.975 

0.956 

0.949 

0.981 

0.975 

0.010 

0.890 

0.905 

0.932 

0.961 

0.941 

0.932 

0.972 

0.963 

0.030 

0.823 

0.848 

0.880 

0.920 

0.914 

0.904 

0.957 

0.939 

0.050 

0.790 

0.817 

0.855 

0.891 

0.889 

0.878 

0.944 

0.925 

0.100 

0.745 

0.779 

0.823 

0.846 

0.860 

0.846 

0.925 

0.910 

0.200 

0.700 

0.750 

0.796 

0.793 

0.827 

0.812 

0.909 

0.891 

0.300 

0.673 

0.738 

0.783 

0.769 

0.807 

0.792 

0.903 

0.889 

0.500 

0.638 

0.731 

0.773 

0.765 

0.779 

0.766 

0.890 

0.884 

0.700 

0.618 

0.734 

0.789 

0.772 

0.761 

0.751 

0.874 

0.879 

1.000 

0.593 

0.752 

0.829 

0.786 

0.742 

0.737 

0.845 

0.877 

2  000 

1  040 

3.000 

l!l64 

1.402 

100 


^  Linhart 
•  Noyea 
X  Harned 
o  Getman 


\ 


k 


log.  Concentration 
Fig.  123 


ELECTROMOTIVE  FORCE  497 

other  similar  data.*  Up  to  the  present  time,  no  satisfactory 
explanation  of  these  abnormalities  has  been  advanced.  The  fact 
that  there  is  a  marked  difference  between  the  values  of  ionization, 
as  calculated  from  freezing-point  and  conductance  measure- 
ments, on  the  one  hand,  and  as  computed  from  electromotive 
force  measurements,  on  the  other,  has  led  several  investigators, 
notably  Ghosh,  f  to  abandon  the  theory  of  Arrhenius  entirely 
and  to  adopt  the  view  that  ionization  is  practically  complete  in 
solutions  of  all  strong  electrolytes^ 

The  theory  of  Ghosh  has  recently  been  critically  analyzed  1$ 
Kendall,  |  who  concludes  that  certain  of  the  fundamental  postu- 
lates upon  which  the  theory  is  based  must  either  be  modified  or 
rejected,  and  that  while  the  theory  of  Arrhenius  undoubtedly 
stands  in  need  of  modification,  no  case  has  yet  been  made  out  for 
its  abandonment.  Therefore,  until  some  alteration  of  the  older 
ionic  theory  is  proposed,  which  will  satisfactorily  explain  the 
abnormal  behavior  of  strong  electrolytes,  we  shall  continue  to 
assume,  that  the  conductance  ratio  affords  a  true  measure  of  the 
degree  of  ionization. 

Heat  of  Ionization.  If  the  difference  of  potential  between  a 
metal  and  the  solution  of  one  of  its  salts  is  known,  it  is  possible 
to  calculate  the  heat  of  ionization  of  the  metal  by  means  of  the 
Gibbs-Helmholtz  equation, 

Q.  dE 

E=nF+TdT1 

provided  the  variation  of  the  electromotive  force  with  tem- 
perature is  known.  Solving  the  equation  for  Qv,  the  heat 
evolved  when  one  mol  of  ions  is  formed  at  the  electrode,  we 
have 


F.  (53) 

For  example,  the  potential  of  zinc  against  a  molar  solution  of 
zinc  chloride  is  —  0.497  volt,  at  25°,  and  the  temperature  coefficient 

*  Linhart,  Jour.  Am.  Chem.  Soc.,  41,  1178  (1919).  Earned,  Ibid.  38, 
1990  (1916).  Getman,  Ibid.  42,  1556  (1920). 

t  Jour.  Chem.  Soc.,  113,  449,  627,  707,  790  (1918);  117,  823,  1390  (1920). 
Trans.  Faraday  Soc.,  15,  148  (1919).  Milner,  Phil.  Mag.  35,  214,  354  (1918). 
Sutherland,  Ibid.  3,  161  (1902).  Bjerrum,  Zeit.  Elektrochem.  24,  321  (1918). 

|  Jour.  Am.  Chem.  Soc.,  44,  717  (1922). 


498 


THEORETICAL  CHEMISTRY 


of  electromotive  force  is  0.000664  volt  per  degree.     Substituting  in 
the  equation,  we  have 

Q.  =  [-  0.497  -  (273  +  25)  X  0.000664]  (2  X  96,540  X  0.2394) 
orQ,  =  32,120  calories. 


That  is,  the  heat  of  the  reaction, 

Zn  +  2  (+) 


Zn", 


is  32, 120  calories  per  mol  of  zinc. 

Gas  Cells.  It  is  interesting  to  note,  that  gases  may  function 
as  electrodes,  in  much  the  same  way  as  metals  or  amalgams.  Gas 
electrodes  are  usually  prepared  by  partially  immersing  strips  of 
platinized  platinum  in  a  solution  of  a  suitable  electrolyte,  and 

bubbling  the  gas  through  the  solution  until 

a  constant  difference  of  potential  is  estab- 
lished between  it  and  the  electrode.  A 
very  satisfactory  form  of  gas  electrode  is 
shown  in  Fig.  124. 

Reference  has  already  been  made  to  the 
hydrogen  electrode  in  connection  with  the 
measurement  of  single  electrode  potentials. 
This  electrode  is  completely  reversible  and 
behaves  like  a  plate  of  metallic  hydrogen, 
the  reaction  at  the  electrode  being  repre- 
sented by  the  equation, 

H2  +  2(-t-)<=±2IT. 

The  amount  of  energy  developed  by  the 
passage  of  a  certain  quantity  of  gas  into 
the  ionic  state  is  precisely  the  quantity  nec- 
essary and  sufficient  to  produce  the  reverse 
action.  This  being  true,  the  metal  of  the 
electrode  can  exert  no  influence  upon  ,the 
electromotive  force. 
A  hydrogen  concentration  cell  can  be  formed  by  connecting 
two  hydrogen  electrodes,  containing  the  gas  at  different  pressures, 
through  an  intermediate  electrolyte.  The  direction  of  the  current 
is  such,  that  the  pressures  on  the  two  sides  of  the  cell  tend  to 
become  equal,  molecular  hydrogen  being  ionized  on  the  high  pres- 
sure side,  and  ionized  hydrogen  being  discharged  on  the  low 


Fig.  124 


ELECTROMOTIVE  FORCE  499 

pressure  side.  The  electromotive  force  of  such  a  cell  can  be  cal- 
culated by  means  of  the  Nernst  equation.  Let  us  consider  a  cell 
composed  of  two  hydrogen  electrodes,  each  at  atmospheric  pres- 
sure, the  H'  ion  concentration  in  each  being  ci,  then, 

RT,      Ci      RT 
E  = 


where  d  is  the  molecular  concentration  of  the  hydrogen  dissolved 
in  the  platinum,  at  gaseous  pressure  pi.  Since  the  hydrogen  is 
present  in  the  form  of  diatomic  molecules,  n  =  2.  If  now  the 
pressure  of  the  gas  at  one  electrode  be  increased  to  p2,  and  the 
corresponding  molecular  concentration  of  the  hydrogen  in  the 
electrode  be  C2,  then  we  shall  have 

v      RT,      Ci      RT.      Cz 
E  =  2F}0^-2F1^ 
or,  since 

Gi  :C2  ::pi  :  p2, 

*  -§£"*£•  (54) 

Equation  (54)  applies  equally  well  to  cells  in  which  two  different 
gases  are  employed.  If  solution  pressures  be  used  in  the  calcu- 
lation of  the  electromotive  force  of  a  gas  cell,  equation  (54)  be- 
comes, 

RT  ,       P\  /f»f»\ 

*  (55) 


where  PI  and  P^  are  the  respective  solution  pressures  of  the 
two  gases.  Since  the  values  of  E,  obtained  by  equations  (54) 
and  (55),  must  be  equal,  we  may  write, 

RT        P±_RT        Pi 

2F10g>2~    F 
Therefore 

and 

l- 

That  is,  the  ratio  of  the  actual  gas  pressures  is  equal  to  the  ratio 
of  the  squares  of  the  corresponding  solution  pressures. 


500  THEORETICAL  CHEMISTRY 

Ionization  of  Water.  An  important  application  of  the  gas 
cell  is  its  use  in  determining  the  degree  of  ionization  of  water. 
If  we  measure  the  electromotive  force  of  the  cell, 

PtH2  -  0.1  m  NaOH  -  0.1  m  HC1  -  PtHz, 

and  determine  the  concentration  of  the  H*  ions  on  one  side,  and  the 
concentration  of  the  OH'  ions  on  the  other  side,  we  can  cal- 
culate the  concentration  of  the  H*  ions  in  the  sodium  hydroxide 
solution.  The  reaction  which  produces  the  current  is  represented 
by  the  equation, 

NaOH  +  HC1  =  NaCl  +  H20, 

or  more  correctly, 

H'  +  OH'  =  H20. 

The  electromotive  force  of  the  above  cell  is  0.646  volt,  at  25°. 
At  the  junction  of  the  two  solutions,  an  electromotive  force  of 
0.0468  volt  is  set  up;  hence,  the  true  electromotive  force  of  the 
cell  is  0.646  +  0.0468  =  0.6928  volt.  The  degree  of  ionization 
of  a  0.1  molar  solution  of  hydrochloric  acid  is  on  =  0.924,  and 
the  degree  of  ionization  of  a  0.1  molar  solution  of  sodium  hy- 
droxide is  0:2  =  0.847.  Introducing  these  values  into  the  Nernst 
equation, 


we  have 

0.6928  =  0.0595  log 


Solving  this  equation,  we  find  GZ  to  be  equal  to  1.66  X  10~13. 
Therefore, 

s  =  CH-  X  COH'  =  1.66  X  10~13  X  0.1  X  0.8^7  =  1.406  X  10~14, 

and 

CH-  =  COH'  =  Vs  =  Vl.406  X  10"14  =  1J87  X  10~7. 

This  value  is  in  excellent  agreement  with  the  values  obtained 
by  various  other  methods  as  shown  in  the  table  on  p.  452. 

When  we  consider  the  exceedingly  small  extent  to  which  water 
is  ionized,  the  close  agreement  between  these  results  is  most 
satisfactory.  The  correctness  of  these  figures  can  be  further 
checked  by  taking  the  values  of  the  degree  of  ionization  of  water 


ELECTROMOTIVE  FORCE  501 

at  two  temperatures,  and  calculating  the  heat  of  the  reaction, 
H*  +  OH7  =  H2O, 

by  means  of  the  equation  of  the  reaction  isochore.  Thus,  accord- 
ing to  Kohlrausch,  the  degree  of  ionization  of  water  is  0.35  X  10~7, 
at  0°,  arid  2.48  X  10~7,  at  50°.  Introducing  these  values  into 
the  equation, 

„      2.3026  R  (log  K2  -  log  Kfi  T^ 

~  W    =    ~  7n m > 

1  2    —    1  1 

and  solving  for  Q,  we  obtain  —  13,810  calories.  This  value  agrees 
well  with  that  found  by  the  direct  measurement  of  the  heat  of 
neutralization  of  completely  ionized  acids  and  bases,  viz.,  —  13,800 
calories. 

The  Hydrogen  Electrode.  The  hydrogen  ion  concentration  of  a 
solution  can  readily  be  calculated  from  the  measured  value  of  the 
electromotive  force  of  a  cell  made  up  according  to  the  following 
scheme : 

H2  -  Solution  -  Saturated  KC1  -  m  KC1,  HgCl  -  Hg. 

The  relation  between  the  unknown  hydrogen  ion  concentration,  CH, 
and  the  electromotive  force  of  the  cell,  at  25°,  is  given  by  the  equa- 
tion of  Nernst,  ^ 

E  =  0.059  log,—  !-  (-  0.283),  (57) 

J 

where  —  0.283  volt  is  the  potential  of  the  normal  calomel  elec- 
trode referred  to  the  hydrogen  electrode  as  zero. 

Therefore,  in  order  to  determine  the  hydrogen  ion  concen- 
tration of  a  solution,  a  hydrogen  electrode  is  immersed  in  the  solu- 
tion, and  the  electromotive  force  of  the  cell,  formed  by  connect- 
ing the  solution  with  a  calomel  electrode,  is  measured.  On  sub- 
stituting the  value  of  the  electromotive  force,  thus  determined,  in 
equation  (57),  the  value  of  the  hydrogen  ion  concentration,  CH, 
can  be  easily  calculated.* 

Hydrogen  ion  concentrations  are  commonly  expressed  by  bac- 
teriologists and  biochemists  in  terms  of  the  numerical  value  of 

log  —  =  pH,  as  originally  suggested  by  Sorensen.  The  advan- 
tage of  this  system  is. that  it  obviates  the  use  of  unwieldy 
figures  in  designating  the  hydrogen  ion  concentration  of 

*  It  is  here  assumed  that  no  boundary  e.m.f.  is  set  up.  Should  such  a 
difference  of  potential  exist,  a  suitable  correction  must  be  applied. 


502  THEORETICAL  CHEMISTRY 

physiological  solutions.  It  must  be  remembered,  however, 
that  physically,  the  number  expressing  the  value  of  pH  has  only 
an  indirect  significance.  The  relation  between  these  two  methods 
of  expressing  hydrogen  ion  concentrations  may  be  illustrated  by 
the  following  example :  If  CH  =  2  X  10~4,  then  according 
to  the  definition,  pH  =  log  l/cH  =  log  1/2  X  lO"4  =  log  5000  = 
3.699. 

Electrometric  Titration.  Since  the  neutralization  of  acids  and 
bases  is  accompanied  by  changes  in  the  concentration  of  hydrogen 
and  hydroxyl  ions,  it  is  apparent  that  the  hydrogen  electrode  can 
be  employed  in  place  of  an  indicator  in  acidimetric  and  alkalimetric 
titrations.  The  first  application  of  the  hydrogen  electrode  to  the 
titration  of  acids  and  bases  was  made  by  Bottger,*  who  showed  that 
the  method  was  applicable  to  titrations  where  ordinary  indicators 
fail  to  give  satisfactory  results,  as  for  example,  in  the  titration  of 
deeply  colored,  or  turbid  solutions.  The  results  of  the  titration 
of  hydrochloric  and  acetic  acids  by  sodium  hydroxide  are  shown 
graphically  in  Fig.  125,  where  the  abscissas  represent  the  number 
of  cubic  centimeters  of  sodium  hydroxide  added  to  the  solution 
from  a  burette,  and  where  the  ordinates  represent  the  correspond- 
ing values  of  the  electromotive  force.  The  scale  of  hydrogen  ion 
concentrations,  at  the  right  of  the  diagram,  is  calculated  by  means 
of  equation  (57).  It  will  be  seen  that,  in  both  cases,  a  sharp  break 
occurs  in  the  curves  when  the  end-point  of  the  titration  is  reached. 
Furthermore,  acetic  acid  shows  an  initial  electromotive  force 
much  greater  than  hydrochloric  acid,  corresponding  to  the  fact 
that  the  hydrogen  ion  concentration  is  much  less.  It  should 
be  noticed  that  the  first  additions  of  alkali  cause  a  marked  de- 
crease in  acidity.  This  shows  the  effect  of  the  acetate  ion  in 
diminishing  the  acidity  of  acetic  acid.  According  to  the  law  of 
mass  action  we  have, 

CHaCOOH  «=>  CHaCOO'  +  IT, 

and  since  the  concentration  of  CHaCOO',  which  was  initially 
small,  due  to  the  slight  ionization  of  the  acid,  is  greatly  increased 
by  the  addition  of  alkali,  it  follows  that  a  corresponding  diminu- 
tion in  the  concentration  of  H*  must  occur.  This  effect  is  of 
sufficient  magnitude  to  cause  the  acidity  to  decrease  very  slowly 
until  the  neutral  point  is  reached,  when  a  sudden  change  to  al- 

*  Zeit.  phys.  Chem.,  24,  253  (1897), 


ELECTROMOTIVE  FORCE 


503 


kaline  reaction  takes  place.  It  can  be  shown,  that  the  middle 
point  of  this  portion  of  the  curve  corresponds  to  the  normal  salt, 
when  both  acid  and  base  are  present  in  exactly  equivalent  amounts. 
It  will  be  observed,  however,  that  this  point  at  which  acid  and 


1.0 
0.9 
0.8 

0.7 
| 

10  -13 

10-12 
10  ~u 

10-io 

10-9 

io-8  ] 

6 

10  •* 
10  -* 

10  -3 

^^f" 

^ 

'/? 

r 

- 

/ 

( 

_ 

Neutral 

Point 

0.6 

CH3CC 

OH 

) 

>^ 

/ 

0,4 

( 

HC1 

) 

0.3 

f  

^-^ 

0  5  10  16  20  25 

c.c.  NaOH 

Fig.    125 

base  are  equivalent  does  not  correspond  to  a  neutral,  but  to  an 
alkaline  solution.  This  is  in  accord  with  the  well-known  fact, 
that  sodium  acetate  solutions  are  hydrolyzed  and  react  alkaline. 

Buffer  Solutions.  If  a  single  drop  of  0.1  m  HC1  be  added  to  a 
liter  of  pure  water,  an  enormous  increase  in  the  hydrogen  ion  con- 
centration will  occur  immediately.  In  like  manner,  the  addition 
of  a  drop  of  0.1  m  KOH  to  a  liter  of  pure  water  will  cause  a  cor- 
respondingly large  increase  in  the  hydroxyl  ion  concentration. 
From  these  facts  it  appears,  that  pure  water  alone  is  incapable  of 


504  THEORETICAL  CHEMISTRY 

neutralizing  even  traces  of  acids  and  bases.  It  is  to  this  lack 
of  reserve  neutralizing  power,  that  the  difficulty  of  preparing  and 
preserving  water  having  a  hydrogen  ion  concentration  of  1  X  10~7 
equivalents  per  liter  is  to  be  ascribed. 

It  has  been  shown  that  any  solution  containing  a  weak  acid 
together  with  one  of  its  salts,  or  a  weak  base  with  one  of  its  salts, 
is  capable  of  neutralizing  both  acids  and  bases,  within  certain 
limits.  This  property  of  certain  solutions  of  resisting  change  in 
hydrogen  ion  concentration,  on  the  addition  of  acid  or  alkali,  is 
known  as  buffer  action,  and  the  solutions  are  called  buffer  solu- 
tions. 

The  explanation  of  buffer  action  is  relatively  simple.  If  to 
a  solution  containing  a  weak  acid,  HA,  and  one  of  its  salts, 
MA,  there  be  added  a  small  amount  of  an  acid  or  base,  the  solu- 
tion will  protect  itself  by  neutralizing  the  added  substance  accord- 
ing to  the  following  reactions  : 

A'  +:  H'  =  HA, 
and 

HA  +  OH'  =  H20  +  A'. 

It  follows,  therefore,  if  the  reserve  acidity  is  to  be  equal  to  the 
reserve  alkalinity,  that  the  concentration  of  A'  must  be  equal  to 
the  concentration  of  HA.  But,  according  to  the  law  of  mass  action, 


hence 

CH-  =  #A. 

That  is,  if  the  reserve  acidity  is  equal  to  the  reserve  alkalinity, 
the  ionization  constant  of  the  acid  will  be  numerically  equal 
to  the  hydrogen  ion  concentration  of  the  solution. 

A  number  of  standard  buffer  solutions  have  been  described  by 
Sorensen,  Palitzsch,  Walpole,  and  Clark  and  Lubs.*  These  solu- 
tions have  been  prepared  with  great  care  and  their  hydrogen  ion 
concentrations  accurately  determined  by  hydrogen  electrode 
measurements.  These  standard  buffer  solutions  are  very  useful 
in  the  preparation  of  solutions  of  definite  hydrogen  ion  concen- 
trations. Thus,  if  it  is  desired  to  titrate  a  solution  to  some 
particular  pH  value,  we  select  from  a  table  of  standard  buffer 

*  See  "  Determination  of  Hydrogen  Ions,"  Clark,  pp.  69  to  83  inch 


ELECTROMOTIVE  FORCE  505 

solutions,  a  formula  which  will  give  a  solution  of  the  desired 
hydrogen  ion  concentration.  The  carefully  prepared  buffer 
solution  is  introduced  into  an  electrode  vessel,  and  the  solu- 
tion to  be  titrated  into  a  second  similar  vessel,  while  con- 
nection between  the  two  solutions  is  established  through  an 
intermediate  saturated  solution  of  potassium  chloride.  A  hy- 
drogen electrode  is  immersed  in  each  of  the  two  solutions,  and 
the  cell  thus  formed  is  included  in  a  circuit  containing  a  galva- 
nometer and  appropriate  keys.  A  small  portion  of  the  titrating 
solution  is  now  added  from  a  burette,  and  the  deflection  of  the 
galvanometer  noted.  The  titration  is  continued  in  this  manner 
until,  upon  closing  the  circuit,  the  galvanometer  shows  no  de- 
flection. When  this  condition  obtains,  the  two  electrodes  are 
manifestly  at  the  same  potential,  and  therefore,  the  two  solutions 
must  have  identical  hydrogen  ion  concentrations.  The  accuracy 
of  the  pH  value  attainable  by  this  method  is  entirely  dependent 
upon  the  accuracy  of  the  pH  value  of  the  standard  buffer  solution.* 

Storage  Cells  or  Accumulators.  Storage  cells  or  accumulators, 
as  the  name  implies,  are  devices  for  the  storage  of  electrical  en- 
ergy in  the  form  of  chemical  energy.  Any  reversible  cell  may  be 
employed  as  an  accumulator.  Thus,  the  oxygen-hydrogen  cell, 

Pto2  —  Solution  of  Sulphuric  Acid  —  PtH2, 

may  be  used  as  an  accumulator,  if  the  gases  resulting  from  the 
electrolysis  of  water  are  collected  at  the  electrodes,  and  then  used 
to  produce  a  current.  Practically,  however,  only  two  types  of 
such  secondary  cells  are  in  use  to-day.  These  are,  the  almost 
universally  used  lead  accumulator,  and  the  less  efficient  nickel- 
iron  accumulator.  Only  the  former  of  these  will  be  considered 
here. 

If  two  lead  plates  are  immersed  in  a  20  per  cent  solution 
of  sulphuric  acid,  a  minute  amount  of  lead  sulphate  will  be 
formed  on  the  surface  of  each  plate.  If  now  a  current  of 
electricity  is  passed  through  the  solution,  the  lead  sulphate  on  the 
cathode  will  undergo  reduction  to  metallic  lead,  and  the  lead  sul- 
phate on  the  anode  will  be  oxidized  to  lead  peroxide.  In  this 
manner  we  form  the  cell, 

Pb  -  20  per  cent  Sol.  H2SO4  -  Pb02. 

*  Buffer  solutions  are  now  widely  used  as  standards  in  color  titrations, 
inasmuch  as  different  indicators  show  characteristic  colors  at  different  hy- 
drogen ion  concentrations  (See  Clark,  loc.  cit.) 


506  THEORETICAL  CHEMISTRY 

The  electromotive  force  of  this  cell  is  about  2  volts.  The  amounts 
of  lead  and  lead  peroxide  produced  in  this  way  are  so  small  that 
the  cell  can  furnish  only  a  very  small  amount  of  electrical  energy. 
In  order  to  increase  its  capacity,  the  electrodes  should  be  given 
as  large  an  amount  of  surface  as  possible.  This  may  be  brought 
about  by  the  method  of  Plante,  in  which  a  solution  of  sulphuric 
acid  is  electrolyzed,  first  in  one  direction  and  then  in  the  other,  thus 
causing  the  plates  to  become  spongy;  or  by  the  method  of  Faure, 
in  which  a  lead  "  grid  "  charged  with  a  paste  of  lead  oxide  and  red 
lead,  is  introduced  into  a  similar  solution,  and  a  current  passed 
until  spongy  lead  is  formed  at  the  cathode  and  lead  peroxide  is 
formed  at  the  anode.  If  the  charging  circuit  is  then  broken  and 
the  two  electrodes  are  connected  by  a  wire,  a  current  will  flow  from 
the  peroxide  plate  to  the  lead  plate,  while  lead  sulphate  is  slowly 
formed  at  each.  When  the  accumulator  is  charged,  the  lead 
sulphate  on  the  negative  electrode  is  reduced  to  metallic  lead,  the 
reaction  being  represented  by  the  following  equation: 

PbS04  +  2  (-)  =  Pb"  +  SO/'. 

At  the  positive  electrode,  SO*"  ions  are  liberated  which  react 
with  the  lead  sulphate  and  the  water  of  the  electrolyte,  in  the 
following  manner: 

PbSO4  +  2  H2O  +  S04"  +  2  (+)  =  PbO2  +  4  H"  -f  2  S04". 

When  the  cell  is  discharged,  S04"  ions  are  liberated  at  the  lead 
plate  and  lead  sulphate  is  formed  as  shown  by  the  equation, 
Pb  +  SO,"  +  2  (+)  =  PbS04. 

At  the  peroxide  plate,  H*  ions  are  discharged  which,  in  the  presence 
of  the  electrolyte,  convert  the  lead  peroxide  into  lead  sulphate, 
according  to  the  equation, 

Pb02  +  2  IT  +  H2S04  +  2 (-)  =  PbS04  +  2  H20. 

Combining  the  foregoing  equations,  we  obtain  the  following  single 
equation,  summarizing  the  chemical  changes  involved  in  the  pro- 
duction of  the  current: 

PbO2  +  Pb  +  2  H2S04  <=»  2  PbS04  +  2  H2O. 

The  upper  arrow  represents  the  reaction  on  discharging,  while 
the  lower  arrow  represents  the  reaction  on  charging.  The 
electromotive  force  of  the  storage  cell  is  approximately  2  volts. 
It  is  not  completely  reversible,  but  under  favorable  conditions 
its  efficiency  may  be  as  high  as  90  per  cent;  that  is,  90  per  cent 


ELECTROMOTIVE  FORCE  507 

of  the  electrical  energy  supplied  to  it  in  charging,  can  be  recovered 
on  discharging.* 

REFERENCES 

Text-book  of  Electrochemistry,    Le   Blanc.     (Translated   by  Whitney  and 

Brown.     Chapter  VII.) 
Electrochemistry,  Lehfeldt.     Chapter  III. 

A  System  of  Physical  Chemistry,  Lewis.     Vol.  II,  Chapters  VII  and  XI. 
Applied  Electrochemistry.     Allmand.     Chapter  VIII. 
Determination  of  Hydrogen  Ions.     Clark.     Chapters  VIII  to  XVII  incl. 
Thermodynamics  and  Chemistry.     Macdougall.     Chapter  XVII. 

PROBLEMS 

1.  Calculate  the  heat  of  amalgamation  of  cadmium  at  0°  from  the 
following  data:  — the  electromotive  force  of  a  cell  made  up  of  a  1-per 
cent  cadmium  amalgam  in  a  solution  of  cadmium  sulphate  is  0.06836  volt 
at  0°,  and  0.0735  volt  at  24°.45.  Ans.  510  cal.  per  mol. 

2.  The  electromotive  force  of  the  cell 

Pb  -  0.01  m  Pb(N03)2  -  sat.  NH4N03  -  m  KC1,  HgCl  -  Hg 

is  -  0.469  volt  at  25°.  The  lead  nitrate  is  62  per  cent  ionized.  What 
is  the  potential  of  lead  against  a  solution  containing  1  mol  of  Pb  ions 
per  liter,  referred  to  the  calomel  electrodes  as  zero? 

Ans.  -  0.405  volt. 

3.  Calculate  the  electromotive  force  of  the  cell 

Cu  amalgam  (a)  —  Solution  CuS04  —  Cu  amalgam  (b), 

at  20°.8,  having  given  that  the  concentrations  of  the  amalgams  (a)  and 
(b)  are  0.0004472  and  0.00016645  respectively.  Ans.  0.0125  volt. 

4.  Calculate  the  electromotive  force  of  the  cell 

Cu  -  m  CuS04  -  0.01  m  CuS04  -  Cu, 

at  25°,  having  given  the  following  values  for  the  degree  of  ionization  of 
the  two  solutions: — for  m  copper  sulphate,  a  =  0.21,  and  for  0.01  m 
copper  sulphate,  «  =  0.61.  The  electromotive  force  at  the  junction  of 
the  two  solutions  may  be  neglected.  Ans.  0.0458  volt. 

5.  At  25°  the  electromotive  force  of  the  cell 

Zn  -  0.5  m  ZnS04  -  0.05  m  ZnS04  -  Zn 

*  For  a  thorough  treatment  of  the  theory  of  the  lead  accumulator  the 
student  is  recommended  to  consult  "The  Theory  of  the  Lead  Accumu- 
lator," by  F.  Dolezalek,  translated  by  C.  L.  von  Ende. 

For  a  detailed  account  of  the  primary  cells  in  common  use  the  reader  will 
find  Carhart's  "  Primary  Batteries  "  most  satisfactory. 


508  THEORETICAL  CHEMISTRY 


is  CbfciS^Sf'fc  Neglecting  the  potential  developed  at  the  junction  of  the 
solutions,  and  assuming  the  dilute  solution  of  zinc  sulphate  to  be  ionized 
to  the  extent  of  35  per  cent,  find  the  degree  of  ionization  of  the  concen- 
trated solution.  Ans.  0.142. 

6.  What  is  the  electromotive  force  of  the  cell 

Zn  -  0.1  m  ZnS04  -  0.01  m  ZnS04  -  Zn, 
at  18°?    For  ZnS04,—  4~  =  0.601,  for  0.1  molar  ZnS04,  «  =  0.39,  and 

U  -j-  V 

for  0.01  molar  ZnS04,  «  =  0.63.  Ans.  0.078  volt. 

7.  The  electromotive  force  of  the  cell 

Ag  -  0.001  m  AgN03  -  m  KN03  -  m  KI,  Agl  -  Ag 

is  0.22  volt  at  18°.  A  molar  solution  of  KI  is  78  per  cent  ionized,  and 
a  0.001  molar  solution  of  AgN03  is  98  per  cent  ionized.  Calculate  the 
solubility  of  Agl.  Ans.  3.53  X  10~3  mols  per  liter. 

8.  The  potential  of  zinc  against  a  solution  containing  one  mol  of  Zn 
ions  per  liter  is  0.493  volt,  at  18°.    Assuming  complete  dissociation, 
calculate  the  solution  pressure  of  zinc  in  atmospheres. 

9.  The  electromotive  force  of  the  Daniell  cell 

Cu  -  CuS04  -  ZnS04  -  Zn 

is  1.0960  volt  at  0°,  and  1.0961  volt  at  3°.  Calculate  the  heat  of  the 
reaction  taking  place  in  the  cell.  Ans.  50250  cal. 

10.  Calculate  the  electromotive  force  of  the  cell 

PtH2  -  0.1  m  KOH  -  m  HC1  -  Ptm, 

at  25°  having  given  that  0.1  molar  KOH  is  85  per  cent  ionized  and  molar 
HC1  i  70  per  cent  ionized.  Ans.  0.757  volt. 

11.  Calculate  the  liquid  junction  potentials  for  each  of  the  following 
pairs  of  solutions  at  25°  :  — 

(a)O.lmHCl,  O.lmKCl, 

(b)  0.05  mHCl,  O.lm  NaCl, 

(c)  0.1  m  KN03,  0.1  m  AgN08. 

12.  The  electromotive  force  of  the  cell 


(Pt) 


0.0337  m  T1(N03)8 
0.0216  m  T1(N03) 
0.42  m  HN03 


Normal  hydrogen  electrode,  is  1.2008  volt  at  25°.  On  the  assumption 
that  the  concentrations  of  the  thallous  and  thallic  ions  are  proportional 
to  the  concentrations  of  the  corresponding  nitrates,  and  regarding  the 
potential  of  the  hydrogen  electrode  as  zero,  calculate  the  normal  potential 


ELECTROMOTIVE  FORCE  509 

of  the  thallic-thallous  ion  electrode.     (The  nitric  acid  is  added  to  prevent 
hydrolysis  of  the  thallic  salt.) 

13.   The  electromotive  force  of  the  cell 


(Pt) 


0.1  m  FeCl3 
0.1  m  FeCl2 
0.1  m  HC1 


m  KC1,  HgCl  Hg, 


is  0.375  volt  at  25°.  The  normal  potential  of  the  ferric-ferrous  electrode 
is  0.430,  calculate  the  value  of  the  ratio  CFe'"/cFe",  on  the  assumption 
that  the  concentrations  of  the  ferrous  and  ferric  ions  are  proportional 
to  the  concentrations  of  the  corresponding  chlorides. 

14.  The  electromotive  force  of  the  cell 

Hg  -  HgCl  -  0.01  m  NaCl  -  0.1  m  KC1  -  HgCl  -  Hg, 

is  0.05403  volt  at  25°,  due  allowance  having  been  made  for  the  junction 
potential.  The  electromotive  force  of  the  cell 

Hg  -  HgCl  -  0.001  m  NaCl  -  0.1  m  KC1  -  HgCl  -  Hg, 

is  0.11200  volt  at  25°,  correction  for  the  junction  potential  having  been 
applied.  The  degree  of  ionization  of  0.001  m  NaCl  is  0.976.  Calculate 
the  activity  coefficient  of  0.01  NaCl. 

15.  Calculate  the  electromotive  force  of  the  cell 

K  (amalgam)  -0.01  m  KC1,  AgCl,  -  Ag- AgCl,  0.1  m  KC1-K  (amalgam). 
K  16.  The  temperature  coefficient  of  the  cell 

Pb  -  PbCl2,  0.1  m  KC1  0.1  m  KC1,  HgCl  -  Hg, 

is  0.00022  and  its  mean  electromotive  force  at  20°  is  0.5382  volt.  Cal- 
culate the  heat  of  the  reaction 

Pb  +  2  HgCl  -H^PbCl2  +  2  Hg. 
/ 

/17.  )The  electromotive  force  of  the  cell 

^H2  -  m/32  C6H5NH2.HC1  -  sat'd  NH4N03  -  m  KC1,  HgCl  -  Hg, 

is  0.4655  volt  at  25°,  the  current  within  the  cell  passing  from  left  to  right. 
Calculate  the  percentage  hydrolysis  of  aniline  hydrochloride. 

18.  The  electromotive  force  of  the  cell 

H2  -  solution  CH3COOH  -  m  KC1,  HgCl  -  Hg, 

is  0.511  at  25°.  Calculate  the  hydrogen  ion  concentration  in  the  solu- 
tion of  acetic  acid.  Express  the  hydrogen  ion  concentration  in  terms  of 
PH. 

19.  A  solution  containing  50  cc.  m/5  KH2P04,  5.70  cc.  m/5  NaOH 
and  diluted  to  200  cc.  with  pure  water  has  been  found  to  have  a  pn  value 


510  THEORETICAL  CHEMISTRY 

of  6.0.  When  a  hydrogen  electrode  is  immersed  in  this  solution  and 
connection  is  established  with  a  normal  calomel  electrode,  what  electro- 
motive force  will  result? 

20.  From  the  electrometric  titration  curve  of  acetic  acid  plotted  in 
Fig.  125,  calculate  the  degree  of  hydrolysis  and  also  the  ionization  con- 
stant of  acetic  acid. 


CHAPTER  XVIII 
ELECTROLYSIS  AND   POLARIZATION 

Polarization.  If  a  difference  of  potential  of  about  1  volt  is 
applied  to  two  platinum  electrodes,  immersed  in  a  concentrated 
solution  of  hydrochloric  acid,  it  will  be  found,  that  the  current 
which  passes  at  first,  will  steadily  diminish  and  ultimately  will 
become  zero.  The  cessation  of  the  current  has  been  shown  to  be 
due  to  the  accumulation  of  hydrogen  on  the  cathode,  and  chlorine 
on  the  anode,  these  two  gases  setting  up  an  opposing  electro- 
motive force,  called  the  electromotive  force  of  polarization.  If 
the  applied  electromotive  for.ce  be  increased  to  1.5  volts, 
the  counter  electromotive  force  will  no  longer  be  sufficient  to 
reduce  the  current  to  zero.  In  fact,  at  any  voltage  above  1.35 
volts  a  continuous  current  passes;  this  is  termed  the  decomposition 
potential  of  hydrochloric  acid. 

At  all  voltages  above  the  decomposition  potential,  the  current, 
C,  may  be  calculated  by  means  of  the  formula, 

E-e  =  CR,  (1) 

where  E  is  the  applied  electromotive  force,  e  the  counter  electro- 
motive force,  and  R  the  resistance  of  the  electrolyte.  As  the 
applied  electromotive  force  and  the  current  increase,  the  polar- 
ization increases,  since  the  gases  are  liberated  under  a  pressure 
greater  than  that  of  the  atmosphere.  Since,  however,  the  gases 
escape  from  the  solution,  the  value  of  e  can  never  become  equal 
to  E. 

The  decomposition  potential  of  an  electrolyte  can  be  deter- 
mined in  either  of  two  different  ways;  (1)  by  gradually  raising 
the  applied  electromotive  force,  E,  until  it  exceeds  e,  when  the 
current  will  suddenly  increase;  or  (2)  by  charging  the  electrodes 
up  to  atmospheric  pressure  by  means  of  an  electromotive  force 
greater  than  e}  and  after  breaking  the  external  circuit,  measur- 
ing the  counter  electromotive  force. 

The  arrangement  of  apparatus  for  the  measurement  of  the 

511 


512 


THEORETICAL  CHEMISTRY 


electromotive  force  of  polarization,  as  suggested  by  Le  Blanc,* 
is  indicated  in  Fig.  126.  A  is  the  electrolytic  cell  in  which  polar- 
ization occurs,  B  is  the  source  of  external  electromotive  force,  C  is 
a  capillary  electrometer,  D  is  a  source  of  variable  potential,  and  E 

, 1 1          is  one  prong  of  an  electrically- 

J-^          f\\r     driven  tuning  fork  which  serves 
d  ~      )         vJ      to   make  and  break  contact  in 
vtx  rapid  alternation  with  the  points, 

|  J-D     F  and  G.     When  the  tuning  fork 

r/\Q  makes  contact  at  F,  the  current 

j  \ flows  through  A,  polarizing  the 
electrodes;  when  contact  is  made 
at  G,  the  counter  electromotive 


B— 


Fig.  126 


force,  due  to  polarization,  causes  a  current  to  flow  through  D 
and  C.  The  counter  electromotive  force  is  balanced  by  varying 
D,  until  no  current  flows  through 
C.  The  potential  of  D  is  then 
equal  to  the  electromotive  force 
of  polarization. 

Just  as  the  electromotive  force 
of  a  galvanic  cell  is  due  to 
the  combined  action  of  several 
differences  of  potential,  so 
also  the  electromotive  force  of 
polarization  is  due  to  the  in- 
dividual differences  of  poten- 
tial located  at  the  electrodes. 
The  method  employed  for  the 
measurement  of  polarization  at  a 
single  electrode  was  devised  by 
Fuchs,  and  is  illustrated  diagram- 
matically  in  Fig.  127.  The  two  electrodes,  A  and  B,  and  the  side 
tube  of  the  normal  calomel  electrode,  C,  are  immersed  in  the 
vessel  containing  the  electrolyte.  D  is  the  source  of  external 
electromotive  force,  E  is  a  capillary  electrometer,  and  F  is  a  source 
of  variable  potential.  Before  closing  the  external  circuit,  DAB,  the 
potential  of  the  electrode,  B,  against  the  solution  is  measured.  Then 
the  circuit,  DAB,  is  closed,  thereby  polarizing  the  electrode  B. 


Fig.  127 


Zeit.  phys.  Chem.,  5,  469  (1890). 


ELECTROLYSIS  AND  POLARIZATION  513 

The  external  circuit  is  now  broken,  and  the  potential  of  B  against 
the  solution  remeasured.  The  difference  between  the  final  and 
initial  values  of  the  electrode  potential,  gives  the  value  of  the 
polarization  at  B.  In  like  manner,  the  polarization  at  the  other 
electrode  can  be  measured. 

The  small  amount  of  electricity  which  is  necessary  to  polar- 
ize an  electrode  is  termed  the  polarization  capacity  of  the 
electrode.  This  factor  is  dependent  upon  the  extent  of  surface  of 
the  electrode,  and  also  upon  the  nature  of  the  metal  of  which  it  is 
formed.  The  polarization  capacity  of  palladium  is  greater  than 
that  of  platinum,  for  electrodes  of  equal  surface,  when  hydrogen 
is  liberated  on  each.  Palladium  dissolves  hydrogen  more  readily 
than  platinum,  and  consequently  a  larger  amount  of  hydrogen, 
and  a  correspondingly  greater  quantity  of  electricity  will  be 
required  to  bring  the  hydrogen  dissolved  in  the  palladium  up  to 
the  same  pressure  as  that  of  the  hydrogen  dissolved  in  the 
platinum.  If,  through  the  processes  of  solution  or  diffusion,  or 
through  chemical  action,  the  substance  which  causes  the  polariz- 
ation is  removed,  the  electrode  is  said  to  be  depolarized.  Thus, 
when  a  reducing  agent,  such  as  ferrous  chloride,  is  electrolyzed, 
the  oxygen  liberated  at  the  anode  immediately  combines  with 
the  electrolyte,  forming  ferric  chloride,  and  preventing  polar- 
ization of  the  electrode. 

If  water  be  electrolyzed  between  platinum  electrodes,  the  cath- 
ode becomes  saturated  with  hydrogen,  and  the  anode  with  oxygen, 
until,  when  the  electromotive  force  of  polarization  becomes  equal 
to  that  of  the  external  circuit,  the  current  ceases.  The  two 
gases,  hydrogen  and  oxygen,  are  soluble,  however,  and  conse- 
quently diffuse  away  from  the  electrodes,  either  escaping  from 
the  solution,  or  recombining  to  form  water.  In  order  to  compen- 
sate for  this  continuous  loss  of  gas  at  the  electrodes,  a  small  cur- 
rent continues  to  flow,  thus  maintaining  the  initial  electromotive 
force  constant.  This  small  current  is  termed  the  residual  current. 
If  oxygen  is  bubbled  over  the  surface  of  the  cathode  during  elec- 
trolysis, the  hydrogen  is  removed  as  rapidly  as  it  is  liberated. 
Such  an  electrode,  on  which  no  new  substance  is  formed  during 
electrolysis,  is  called  an  unpolarizable  electrode. 

Decomposition  Potentials.  The  decomposition  potential  of  an 
electrolyte  can  be  determined,  as  has  already  been  pointed  out, 
by  immersing  two  platinum  electrodes  in  the  solution  and  connect- 


514 


THEORETICAL  CHEMISTRY 


ing  with  a  source  of  electricity,  the  electromotive  force  of  which 
can  be  varied  at  will.  The  voltage  is  gradually  increased  and  the 
corresponding  current  is  observed.  It  is  found  that  the  current 
increases  at  first,  and  then  drops  almost  to  zero  every  time  the 
voltage  is  raised,  until  the  decomposition  potential  is  ultimately 


Electromotive  Force 
Fig.  128 

reached.  Beyond  this  point  the  current  has  been  found  to 
be  directly  proportional  to  the  electromotive  force.  If  the 
applied  electromotive  forces  are  plotted  as  abscissae,  and  the 
corresponding  currents  as  ordinates,  we  obtain  curves  of  the 
form  shown  in  Fig.  128.  Some  of  the  decomposition  potentials 
of  molar  solutions,  as  determined  by  Le  Blanc,*  are  given  in  the 
accompanying  tables. 


DECOMPOSITION  POTENTIALS 

SALTS 


Salt. 

Decomp. 
Potential. 

Salt. 

Decomp. 
Potential. 

ZnS04.  .  . 

Volts. 

2  35 

Cd(NO3)2.., 

Volts. 

1.98 

ZnBr2 

1  80 

CdSO4 

2  03 

NiSO4 

2  09 

CdCl2 

1.88 

NiCl2  

1  85 

CoSO4  

1.92 

Pb(NO3)2.. 

1  52 

CoCl2  

1.78 

AgNOs  

0.70 

Zeit.  phys.  Chem.,  8,  299  (1891), 


ELECTROLYSIS  AND   POLARIZATION 


515 


ACIDS 


Acid. 


Volts. 

H2SO4 67 

HNO3 69 

H3P04 70 

CH2C1.COOH .72 

CHC12.COOH 66 

CH2(COOH)2 69 

HC104 65 

HC1 31 

(COOH)2 0.95 

HBr 0.94 

HI 0.52 

BASES 

Decomp. 
Potential. 

Volts. 

NaOH 1.69 

KOH 1.67 

NH4OH..  1.74 


Decomp. 
Potential. 


It  will  be  observed,  that  while  there  is  considerable  variation  in 
the  decomposition  potentials  of  salts,  there  is  very  little  variation 
in  the  decomposition  potentials  of  acids  and  bases.  There  is  a 
maximum  value  of  about  1.70  volts  to  which  many  acids  and 
bases  closely  approximate.  It  is  found  that  all  acids  and  bases, 
which  decompose  at  1.70  volts,  give  off  hydrogen  and  oxygen  at 
the  electrodes.  Those  acids  and  bases  which  decompose  at 
potentials  less  than  the  maximum,  do  not  liberate  hydrogen  and 
oxygen.  When  their  solutions  are  sufficiently  diluted,  however, 
hydrogen  and  oxygen  are  evolved,  and  the  decomposition  potential 
rises  to  the  maximum  value.  Thus,  Le  Blanc  found  the  follow- 
ing values  for  the  decomposition  potential  of  different  dilutions  of 
hydrochloric  acid. 

DECOMPOSITION  POTENTIALS  OF  HC1 


Concentration. 


2mHCl 
ImHCl 
ImHCl 


Decomp. 
Potential. 


Volta. 
.26 
.34 
.41 
.62 


516  THEORETICAL  CHEMISTRY 

When  2  m  hydrochloric  acid  is  electrolyzed,  hydrogen  and 
chlorine  are  given  off  at  the  electrodes,  whereas,  when  the  concen- 
tration of  the  acid  is  reduced  to  1/32  m,  hydrogen  and  oxygen 
are  the  products  of  electrolysis,  and  the  decomposition  potential 
increases  to  1.70  volts.  It  is  found  that  the  values  of  the  decom- 
position potentials  vary  slightly  with  the  nature  of  the  electrodes. 
The  above  values  refer  to  platinum  electrodes. 

The  Theory  of  Polarization.  Our  knowledge  of  the  processes 
taking  place  'at  the  electrodes,  during  electrolysis,  is  largely  due 
to  the  investigations  of  Le  Blanc.  He  determined  the  electro- 
motive force  of  polarization  at  each  electrode,  varying  the  external 
electromotive'  force  from  zero  up  to  the  decomposition  potential 
of  the  solution.  When  the  decomposition  value  was  reached,  the 
potential  of  the  electrode  against  the  solution  was  found  to  be  the 
same  as  the  difference  of  potential  between  the  solution  and  the 
element  liberated  at  the  electrode.  Thus,  the  decomposition 
potential  of  a  molar  solution  of  zinc  sulphate  was  found  to  be  2.35 
volts,  while  the  corresponding  difference  of  potential  between  the 
electrode  and  the  solution  was  found  to  be  0.493  volt.  If  a  piece 
of  pure  zinc  is  immersed  in  a  molar  solution  of  zinc  sulphate,  the 
difference  of  potential  will  be  found  to  be  0.493  volt,  the  metal 
being  negative  to  the  solution.  It  frequently  happens  that,  before 
the  decomposition  point  of  the  solution  is  reached,  the  electrode 
exhibits  the  potential  of  the  deposited  metal.  For  example,  in  a 
molar  solution  of  silver  nitrate,  the  electrode  acquires  the  potential 
of  pure  silver  in  molar  silver  nitrate,  before  the  decomposition 
value,  0.70  volt,  is  reached.  This  is  due  to  the  fact,  that  the  osmotic 
pressure  of  the  silver  ions  exceeds  the  solution  pressure  of  the 
metal,  resulting  in  the  deposition  of  the  ions  of  the  metal  without 
the  application  of  any  external  electromotive  force. 

When  an  indifferent  electrode,  such  as  platinum,  is  immersed  in 
a  solution  of  a  salt,  a  very  small  amount  of  ionic  deposition  must 
occur,  otherwise,  according  to  the  Nernst  equation,  an  infinite 
electromotive  force  must  be  established.  Thus,  in  the  equation, 

P 

-,  (2) 


if  the  solution  pressure  P  =  0,  it  is  evident,  that  E  =  °o  ,  and  a 
perpetual  motion  must  result.  We  are  thus  forced  to  the  con- 
clusion, that  when;  an  indifferent  electrode  is  immersed  in  a  salt 


ELECTROLYSIS  AND   POLARIZATION  517 

solution,  ions  will  continue  to  separate  upon  it  until  the  tend- 
ency for  the  deposited  metal  to  go  back  into  solution  exactly 
counterbalances  the  tendency  to  separation.  Hence,  the  elec- 
trode will  become  positive  toward  the  solution.  The  magni- 
tude of  this  difference  of  potential  will  be  dependent  upon  the 
amount  of  metal  deposited.  It  is  to  be  noted  that  this  difference 
of  potential  need  not  be  equal  to  that  between  the  massive  metal 
and  the  solution.  If  the  electrodes  be  connected  with  an  external 
source  of  electromotive  force,  the  value  of  which  can  be  varied  at 
will,  and  a  small  electromotive  force  be  applied,  more  metal  will 
separate  on  the  cathode.  This  will  cause  an  increase  in  the  solu- 
tion pressure  P,  tending  to  offset  further  deposition.  A  still 
further  increase  in  the  external  electromotive  force  will  cause  the 
deposition  of  more  metal  and,  as  a  result  of  the  corresponding 
increase  in  P,  further  deposition  at  that  voltage  will  be  pre- 
vented. Ultimately,  when  the  applied  electromotive  force  is 
such  that  P  acquires  its  maximum  value,  equivalent  to  that  of 
the  massive  metal,  continuous  deposition  will  occur.  An  exactly 
analogous  process  takes  place  at  the  anode.  If  a  gas  is  liberated, 
its  concentration  steadily  increases  until  the  maximum  pressure 
is  reached,  when  it  will  escape  from  the  solution.  When  strong 
currents  are  employed,  P  does  not  remain  constant,  as  has  been 
assumed  above,  but  gradually  diminishes,  causing  the  difference 
of  potential  at  the  electrode  to  increase.  From  the  above  con- 
siderations it  becomes  clear  that  a  continuous  decomposition 
of  an  electrolyte  will  take  place  only  when  the  concentrations 
of  the  substances  separating  at  the  electrodes  have  attained 
their  maximum  values.  When  the  decomposition  point  is 
reached,  the  electrode  exhibits  the  potential  characteristic  of 
the  massive  metal.  It  is  evident  from  the  behavior  of  silver 
nitrate,  and  the  salts  of  other  metals,  for  which  p  >  P,  that  the 
maximum  values  of  concentration  at  the  electrodes  need  not 
necessarily  be  attained  simultaneously. 

When  the  products  of  electrolysis  are  gaseous,  the  value  of  the 
decomposition  potential  depends  upon  the  nature  of  the  electrodes. 
Thus,  the  cell, 

PtH2  -  m  H2SO4  -  Pto2 

gives  an  electromotive  force  of  1.07  volts,  if  platinized  platinum 
electrodes  are  used.  If  an  external  electromotive  force  slightly 


518  THEORETICAL  CHEMISTRY 

greater  than  1.07  volts  be  applied  to  this  cell,  in  the  reverse  direc- 
tion, a  continuous  decomposition  of  water  will  take  place,  and 
hydrogen  and  oxygen  will  be  evolved  at  the  electrodes.  If 
on  the  other  hand,  the  platinized  electrodes  are  replaced  by 
electrodes  of  polished  platinum,  the  decomposition  potential 
will  be  found  to  rise  to  1.68  volts.  The  reverse  electromotive 
force  of  polarization,  however,  is  found  to  be  only  1.07  volts. 
That  is,  the  liberation  of  gas  at  a  polished  platinum  electrode  is 
an  irreversible  process.  The  difference  in  the  behavior  of  the 
two  electrodes  is  explicable  when  it  is  remembered,  that  platinum 
is  capable  of  occluding  large  amounts  of  gas.  A  platinized  elec- 
trode absorbs  the  liberated  gas  very  slowly,  and  when  thoroughly 
saturated,  if  it  is  not  entirely  immersed  in  the  solution,  it  grad- 
ually gives  up  the  gas  by  diffusion,  no  bubbles  being  formed. 
Thus,  if  the  external  electromotive  force  be  raised  to  1.07  volts, 
the  system  will  be  in  equilibrium,  while  if  the  applied  electromo- 
tive force  be  greater,  or  less,  than  the  equilibrium  value,  a  current 
will  flow  in  one  direction,  or  the  other,  gas  being  either  liberated 
or  dissolved.  In  other  words,  the  cell  is  completely  reversible. 

Where  polished  platinum  or  gold  electrodes  are  used,  however, 
the  decomposition  potential  is,  as  has  been  stated,  1.68  volts. 
Polished  electrodes  have  relatively  small  absorbing  power. 
Hence,  if  an  electromotive  force  between  1.07  and  1.68  volts  be 
applied,  the  gases  cannot  diffuse  away  from  the  electrode  rapidly 
enough,  and,  when  the  solution  in  the  vicinity  of  the  electrodes 
becomes  saturated  with  gas,  the  current  will  cease  flowing. 
A  very  slow  process  of  diffusion  from  the  solution  into  the  air 
is  constantly  taking  place,  however,  and  this  permits  the  continu- 
ous evolution  of  an  exceedingly  small  amount  of  gas,  while  a  corre- 
spondingly small  current  traverses  the  solution. 

In  order  to  produce  a  steady  electrolysis,  it  is  necessary  to  raise 
the  external  electromotive  force  to  such  an  intensity  that 
bubbles  are  formed  continuously  at  the  surface  of  the  electrodes. 
This  calls  for  the  performance  of  work,  the  amount  of  which  is 
dependent  upon  the  condition  of  the  electrode  surfaces,  the  sur- 
face tension  of  the  solution,  and  various  other  factors.  In  cases 
where  bubbles  are  formed,  a  portion  of  the  available  energy  of 
the  chemical  process  is  not  expended  in  effecting  electrical  separa- 
tion; consequently,  the  reverse  electromotive  force  is  less  than 
the  applied,  and  the  system  is  irreversible. 


ELECTROLYSIS  AND  POLARIZATION 


519 


Overvoltage.  The  reactions  occurring  at  the  electrodes  of  an 
electrolytic  cell  are  catalytically  accelerated  by  the  metal  of  which 
the  electrodes  are  formed.  Thus,  platinized  platinum  is  the 
most  effective  catalyst  for  the  reaction  represented  by  the 

equation, 

H2  +  2(+)<=±2H*. 

Hydrogen  is  liberated  on  platinized  platinum  at  0  volt,  on 
polished  platinum  at  0.09  volt,  and  on  zinc  at  0.70  volt. 
The  electromotive  force  necessary  to  overcome  the  resistance  of 
the  chemical  reaction  at  an  electrode  is  termed  overvoltage. 
Thus,  we  say  that  hydrogen  is  liberated  on  polished  platinum  with 
an  overvoltage  of  0.09  volt,  and  on  zinc  with  an  overvoltage  of 
0.70  volt.  The  following  table  gives  the  overvoltage  necessary 
for  the  liberation  of  hydrogen  and  oxygen  on  electrodes  of  differ- 
ent metals 

ELECTRODE  OVERVOLTAGES 


Hydrogen  Liberation. 

Oxygen  Liberation. 

Metal. 

Overvoltage. 

Metal 

Overvoltage. 

Pt  (platinized)  

0.00 
0.01 
0.08 
0.09 
0.15 
0.21 
0.23 
0.46 
0.53 
0.64 
0.70 
0.78 

Au.  . 

1.75 
1.67 
1.65 
1.65 
1.63 
1.53 
1.48 
1.47 
1.47 
1.36 
1.35 
1.28 

Au 

Pt  (polished)  
Pd.. 

Fe  (in  NaOH)  

Pt  (polished)  

ACT     . 

Cd  

AK 

Ni 

Pb 

Cu... 

Cu.. 

Pd 

Fe       "    '   " 

Sn.. 

Pt  (platinized) 

Pb.  . 

Co 

Zn.. 

Ni  (polished) 

He;  . 

Ni  (spongv) 

Various  theories  as  to  the  cause  of  overvoltage  have  recently 
been  advanced.  As  the  result  of  a  rather  extensive  experimental 
investigation,  Newberry  *  concludes  that  overvoltage  is  deter- 
mined by  three  different  factors,  namely:  (1)  Supersaturation  of 
the  electrode  surface  with  non-electrified  gas  under  very  high 
pressure,  due  to  the  permeability  of  the  metal  to  the  ionized  gas; 
(2)  deficiency  or  excess  of  non-hydrated  ions  in  the  immediate 
neighborhood  of  the  electrodes;  (3)  inductive  action  of  the  es- 
caping ionized  gas  on  the  electrode. 

*  Jour.  Chem.  Soc.,  105,  2419  (1914). 


520  THEORETICAL  CHEMISTRY 

A  general  theory  of  overvoltage  has  been  developed  by  Bennett 
and  Thompson.*  According  to  their  theory,  the  excess  of  back 
electromotive  force  over  the  reversible  electromotive  force,  is 
due  to  the  accumulation,  during  electrolysis,  of  unstable  products 
above  the  equilibrium  concentration.  Among  such  products  are 
active  hydrogen,  HI,  active  oxygen,  Oi  and  MI,  atomic  metal, 
analogous  to  vaporized  metal,  in  the  case  of  metal  overvoltages. 
These  products  are  shown  to  be  more  reactive  than  the  final  prod- 
ucts, and  are  sufficiently  active  to  explain  the  overvoltages  found 
experimentally.  The  authors  claim  for  their  theory  that  it  offers 
an  entirely  satisfactory  explanation  of  the  known  facts  concerning 
overvoltage. 

Maclnnes  and  Adler,  f  as  a  result  of  their  experimental  studies 
on  hydrogen  overvoltage,  conclude  that  it  is  due  primarily  to  a  layer 
of  supersaturated,  dissolved  hydrogen  in  the  electrolyte  surround- 
ing an  electrode.  If  the  electrode  is  capable  of  adsorbing  suffi- 
cient hydrogen  to  form  nuclei  for  bubbles  of  gas,  the  supersatura- 
tion  cannot  rise  to  high  values,  and  the  electrode  will  have  a  low 
overvoltage.  On  the  other  hand,  metals  possessing  small  adsorp- 
tive  powers  exhibit  high  overvoltages.  The  authors  claim  to 
have  furnished  experimental  evidence  in  support  of  their 
theory. 

Primary  Decomposition  of  Water  in  Electrolysis.  The  decom- 
position potential  of  an  electrolyte  giving  off  hydrogen  and  oxygen 
at  the  electrodes,  is  dependent  upon  the  concentrations  of  the  two 
ions,  H*  and  OH',  and  is  independent  of  the  nature  of  the  electro- 
lyte. As  has  already  been  stated,  the  decomposition  potential  of 
all  acids  and  bases,  which  give  off  hydrogen  and  oxygen,  approxi- 
mates to  1.70  volts.  According  to  the  law  of  mass  action,  the 
product  of  the  concentrations  of  the  H*  and  OH'  ions  is  constant, 
and  independent  of  the  other  substances  which  may  be  present. 
Therefore,  it  follows,  that  although  the  values  of  the  single 
potential  differences  may  vary  widely  on  changing  the  sol- 
vent, their  sum  is  invariably  the  same.  Excluding  solu- 
tions of  salts  which  undergo  reduction  by  hydrogen,  and 
solutions  of  chlorides,  bromides,  and  iodides  reducible  by 
oxygen,  the  ions  H*  and  OH',  according  to  Le  Blanc,  are  to 
be  regarded  as  the  sole  factors  in  the  electrolysis  of  solutions,  and 

*  Jour.  Phys.  Chem.,  20,  296  (1916). 
t  Jour.  Am.  Chem.  Soc.,  41,  194  (1919). 


ELECTROLYSIS  AND   POLARIZATION  521 

not  the  ions  of  the  dissolved  electrolyte.  In  other  words,  elec- 
trolysis involves  a  primary  decomposition  of  water. 

The  electrical  conductance  of  the  solution  is  due  to  the  ions  of 
the  electrolyte  together  with  the  ions  of  water,  but  at  the  electrode 
that  process  takes  place  which  involves  the  expenditure  of  the 
minimum  amount  of  energy,  and  this  is,  under  ordinary  conditions, 
the  separation  of  the  H*  and  OH'  ions.  Thus,  when  a  solution 
of  potassium  sulphate  is  electrolyzed  by  a  current  of  moderate 
strength,  it  is  not  rational  to  assume  that  the  K*  and  SO/7  ions 
are  primarily  discharged  at  the  electrodes,  and  that  these  sub- 
sequently react  with  water.  This  may  be  made  clear  by  consider- 
ing the  process  taking  place  at  the  cathode.  According  to  the 
explanation  based  upon  so-called  "secondary  action,"  the  K*  ions 
give  up  their  positive  charges  to  the  electrode,  and  then  react 
with  water  as  indicated  by  the  equation, 

K  +  H*  +  OH'  -» K'  +  OH'  +  H. 

This  explanation  involves  the  transfer  to  the  potassium  atom  of 
the  positive  charge  of  the  H*  ion  of  water;  this  can  only  take  place 
if  the  H'  ion  holds  its  charge  less  tenaciously  than  the  K*  ion. 
Hence,  if  the  H*  ion  parts  with  its  charge  more  readily  than  the 
K'  ion,  the  former  will  be  discharged  primarily  at  the  cathode. 
Similar  reasoning  may  be  employed  to  explain  the  action  at  the 
anode.  Therefore,  in  electrolysis  all  of  the  ions  participate  in 
conducting  the  current  and  collect  around  the  electrodes,  but 
since  the  H*  and  OH'  ions  separate  more  easily,  these  are  dis- 
charged primarily.  With  stronger  currents  it  is  possible  to  cause 
the  separation  of  the  K'  and  SO4"  ions  also,  since  the  number  of 
H*  and  OH'  ions  present  is  too  small  to  carry  all  of  the  current, 
and  the  energy  required  to  discharge  the  ions  of  the  electrolyte  is 
less  than  that  necessary  to  remove  the  small  number  of  residual  H* 
and  OH'  ions.  The  formation  and  decomposition  of  water  are 
reversible  processes,  so  that  no  loss  of  energy  is  involved,  as 
would  be  the  case  if  secondary  actions  occurred. 

Electrolytic  Separation  of  the  Metals.  Freudenberg  *  was  the 
first  to  recognize  the  possibility  of  effecting  the  quantitative 
separation  of  different  metals  by  means  of  graded  electromotive 
forces.  He  showed  that  it  was  only  necessary  to  select  a  salt, 

*  Zeit.  phys.  Chem.,  12,  97  (1893). 


522 


THEORETICAL  CHEMISTRY 


of  each  metal,  whose  respective  decomposition  potentials  differ 
as  widely  as  possible,  and  electrolyze  at  an  electromotive  force 
intermediate  between  these  potentials.  The  salt  having  the  lower 
decomposition  potential  will  decompose  first,  and  when  the  de- 
position of  the  metal  is  complete,  the  current  will  practically 
cease  flowing;  then  if  the  applied  electromotive  force  be  raised 
above  the  decomposition  potential  of  the  second  salt,  the  second 
metal  will  be  deposited.  In  practice  it  is  found  necessary  to 
increase  the  applied  electromotive  force  slightly,  because  of  the 
gradual  decrease  in  the  number  of  ions  of  the  salt  having  the 
lower  decomposition  potential.  The  amount  of  this  increase  may 
be  readily  calculated  from  the  familiar  equation, 

RT 


For  example,  suppose  a  mixture  of  the  nitrates  of  cadmium,  lead 
and  silver  is  subjected  to  electrolysis,  the  decomposition  potentials 
of  the  salts  being  as  follows:  —  Cd(NO3)2  =  1.98  volts,  Pb(NO3)2  = 
1.52  volts,  and  AgNO3  =  0.70  volt.  If  the  applied  electromotive 
force  is  made  a  little  less  than  1  volt  all  of  the  silver  will  be 
deposited;  if  the  electromotive  force  is  raised  to  about  1.6  volts, 
all  of  the  lead  will  be  deposited;  and  finally,  if  the  electro- 
motive force  be  raised  to  about  2  volts  the  cadmium  will  be 
deposited. 

In  the  subjoined  table  are  given  the  separation  potentials  of 
some  of  the  ions,  the  separation  potential  of  the  H*  ion  being  as- 
sumed to  be  equal  to  zero. 


SEPARATION  VALUES  OF  IONS  FOR  MOLAR 
CONCENTRATION 


Ion 

Separation 
Potential 

Ion 

Separation 
Potential 

Ag*.. 

+0  78 

I' 

-052 

cu-.: 

+0  34 

Br' 

-0.94 

H*... 

—0  0 

O". 

—  1.08  (in  acid) 

Pb".... 

—0  17 

cr  

-1.31 

Cd".... 

—0  38 

OH' 

—  1  68  (in  acid) 

Zn"  

-0  74 

OH' 

—0  88  (in  base) 

SO4" 

—  1.9 

ELECTROLYSIS  AND  POLARIZATION  523 

According  to  this  table  the  decomposition  potential  of  water  is 
equal  to  the  sum  of  the  separation  potentials  of  its  ions,  or  1.68 
volts. 

REFERENCES 

Text-book  of  Electrochemistry.     Le  Blanc.     (Translated  by  Whitney  and 

Brown.)     Chapter  VIII. 
Applied  Electrochemistry.     Allmand.     Chapters  IX  and  X. 


CHAPTER  XIX 
PHOTOCHEMISTRY 

Radiant  Energy.  The  visible  portion  of  the  spectrum  is  com- 
prised between  the  extreme  red,  at  one  end,  and  the  extreme  violet 
at  the  other;  the  wave-length  corresponding  to  the  former  is 
approximately  0.7  micron,  while  that  corresponding  to  the  latter 
is  about  0.4  micron.  The  visible  portion  of  the  spectrum,  how- 
ever, is  but  a  small  fraction  of  the  entire  spectrum.  Beyond  the 
red  of  the  visible  spectrum  lies  the  region  of  the  so-called  infra- 
red, comprising  all  wave-lengths  from  0.76  micron  up  to  300  mi- 
crons. Beyond  the  infra-red,  between  300  and  2000  microns,  is 
an  unmeasured  region,  which  is  succeeded  by  the  region  of  elec- 
trical waves,  extending  from  2000  microns  to  an  undetermined 
maximum.  On  the  other  hand,  extending  beyond  the  violet  of 
the  visible  spectrum  is  the  so-called  ultra-violet  or  actinic  region, 
comprising  all  wave-lengths,  between  0.4  micron  and  0.1  micron. 
It  thus  appears  that  heat,  light,  and  electricity  are  all  forms  of 
radiant  energy,  the  only  distinction  between  them  being  a  differ- 
ence in  wave-length.  Very  little  is  known  concerning  radiant 
energy,  and  up  to  the  present  time  all  attempts  to  resolve  it  into 
a  capacity  and  an  intensity  factor  have  failed.  Whatever  may  be 
the  nature  of  this  form  of  energy,  we  know  that  the  effects  pro- 
duced by  it  are  dependent  upon  the  wave-length  of  the  radiation. 

We  have  already  devoted  several  chapters  to  the  consideration 
of  thermochemistry  and  electrochemistry,  and  it  now  remains  to 
study,  very  briefly,  the  connection  between  chemical  energy  and 
that  subdivision  of  radiant  energy  called  light.  This  branch  of 
theoretical  chemistry  is  termed  photochemistry. 

The  ultra-violet,  or  actinic  rays  are  the  most  active  chemically, 
although  light  of  every  wave-length,  including  the  invisible  infra- 
red, is  capable  of  producing  chemical  action.  When  light  falls 
upon  a  substance,  a  portion  of  the  incident  radiation  is  reflected,  a 
portion  is  absorbed,  and  a  portion  is  transmitted.  It  has  been 
shown,  that  only  that  portion  of  the  incident  radiation  which  is 
absorbed  is  effective  in  producing  chemical  change. 

524 


PHOTOCHEMISTRY  525 

Radiant  energy  has  been  shown  by  Lebedew,*  and  also  by 
Nichols  and  Hull,  f  to  exert  a  definite,  though  extremely  small 
pressure.  Thus,  the  pressure  of  solar  radiations  on  the  earth  is 
equivalent  to  that  of  a  column  of  mercury  1.4  X  10~9  mm.  high. 

Source  of  Radiant  Energy.  According  to  the  electromagnetic 
theory  of  light,  the  emission  of  waves  of  light  from  a  material 
source  is  due  to  the  vibrations  of  minute  charged  particles  called 
radiators.  These  radiators,  which  may  be  either  atoms  or  elec- 
trons, give  rise  to  electromagnetic  waves  of  the  same  period  as 
their  own,  that  is,  to  light  waves  of  definite  length.  The  energy 
required  to  produce  these  electromagnetic  waves  is  derived  from 
the  vibrating  system  itself,  and  unless  an  equivalent  amount  of 
energy  is  constantly  supplied  to  the  system,  the  amplitude  of  the 
vibrations  will  steadily  diminish  and  ultimately  cease. 

There  are  two  different  ways  in  which  this  supply  of  energy  can 
be  maintained.  First,  the  temperature  of  the  vibrating  system, 
as  a  whole,  may  be  kept  high.  This  type  of  radiation,  which  is 
maintained  by  purely  physical  means,  is  called  pure  temperature 
radiation.  Every  substance  whose  temperature  is  above  the  abso- 
lute zero  (  —  273°)  gives  rise  to  pure  temperature  radiation.  The 
higher  the  temperature,  the  more  rapid  and  the  more  energetic  the 
atomic  and  electronic  vibrations  become.  With  increase  in  rapid- 
ity of  vibration,  there  results  a  corresponding  diminution  in  wave- 
length, so  that,  as  the  temperature  is  raised,  the  longer  heat  waves 
are  succeeded  by  the  shorter  waves  of  the  visible  region  of  the 
spectrum.  When  a  sufficiently  high  temperature  is  reached,  the 
period  of  vibration  becomes  so  rapid  as  to  cause  the  radiation  of 
waves  corresponding  to  the  entire  range  of  the  visible  spectrum. 
At  higher  temperatures,  the  rate  of  vibration  is  such  that  the 
vibrating  particles  must  of  necessity  possess  extremely  small  mass. 
It  is  commonly  believed  that  under  these  conditions  the  vibration 
is  wholly  electronic. 

The  second  way  in  which  energy  may  be  supplied  to  the  vibrat- 
ing atoms,  or  electrons,  is  by  chemical  or  electrical  means.  The 
general  term  luminescence  has  been  proposed  by  Wiedemann  for 
all  cases  where  luminous  energy  is  derived  from  other  sources  than 
high  temperature.  It  is  to  be  observed,  that  luminescence  is  fre- 
quently exhibited  by  systems  whose  temperatures  are  compara- 

*  Rapp.  pres  au  Congres  de  Physique,  2,  133  (1900). 
f  Phys.  Rev.,  13,  293  (1901). 


526  THEORETICAL   CHEMISTRY 

lively  low.  For  example,  notwithstanding  the  fact  that  the  flame 
resulting  from  the  combustion  of  carbon  disulphide  has  a  temper- 
ature of  only  150°,  it  has  been  found  to  be  capable  of  affecting  the 
photographic  plate.  Pure  temperature  radiation  alone,  at  150°, 
would  correspond  to  long  waves  in  the  infra-red  region  of  the  spec- 
trum and,  as  is  well  known,  such  waves  are  incapable  of  exerting 
appreciable  photographic  action. 

Emission  and  Absorption.  The  relation  between  the  emissive 
and  absorptive  powers  of  different  bodies  was  first  clearly  enun- 
ciated by  Kirchhoff  *  in  1859.  This  law  may  be  stated  as  follows : 
—  Light  of  any  given  wave-length  emitted  by  a  body  can  also  be  ab- 
sorbed by  the  same  body  at  a  lower  temperature.  This  law,  it  will 
be  seen,  offers  a  satisfactory  explanation  of  the  Fraunhofer  lines  in 
the  solar  spectrum.  The  sun  is  surrounded  by  a  gaseous  atmos- 
phere, resulting  from  the  vaporization  of  the  elements  present  in 
the  body  of  the  sun.  Each  element  in  the  cooler  gaseous  envelope, 
according  to  KirchhofFs  law,  absorbs  those  wave-lengths  which  it 
emits  at  the  higher  temperature  of  the  solar  nucleus.  The  result- 
ing dark  lines  of  the  solar  .spectrum  have  enabled  the  astronomer 
to  determine  the  elementary  composition  of  the  sun. 

If  the  emissive  and  absorptive  powers  of  a  body  be  denoted  by 
E  and  A  respectively,  then  according  to  KirchhofFs  law, 

E/A  =  S,  (1) 

where  S  is  a  constant.  When  absorption  is  complete,  A  is  unity 
and  S  =  E.  Under  these  conditions  the  constant,  S,  may  be  de- 
fined as  the  emissivity  of  a  body  which  absorbs  all  of  the  incident 
radiation  and  reflects  none.  Such  a  body  was  called  by  Kirch- 
hoff a  perfectly  black  body.  The  emissivity  of  a  perfectly  black 
body  is  equal  to  the  ratio  of  the  emissive  to  the  absorptive  power 
of  any  body  at  the  same  temperature.  A  familiar  qualitative 
illustration  of  KirchhofFs  law  is  that  afforded  by  the  appearance  of 
a  fragment  of  white  chinaware,  decorated  with  a  dark  pattern, 
when  heated  to  a  high  temperature.  The  dark  parts  of  the  de- 
sign absorb  light,  while  the  white  parts  reflect  it.  On  heating 
the  fragment  to  redness,  the  pattern  will  be  reversed,  the  dark 
portions  of  the  design  appearing  bright,  and  the  white  portions 
dark. 

It  has  been  shown  that  the  law  of  Kirchhoff  is  a  necessary  con- 
*  Ostwald's  Klassiker,  No.  100  (1898). 


PHOTOCHEMISTRY  527 

sequence  of  the  application  of  the  second  law  of  thermodynamics 
to  the  thermal  equilibrium  within  an  enclosure  whose  walls  are 
impervious  to  heat. 

The  Stefan-Boltzmann  Law.  From  a  study  of  the  experi- 
ments of  Dulong  and  Petit  on  the  rate  of  cooling  of  different  bodies, 
Stefan  *  discovered  an  empirical  relation  between  the  total  radia- 
tion of  a  body  and  its  temperature.  Later,  Boltzmann  f  derived 
the  same  relation  thermodynamically  and  showed  that  instead  of 
being  general,  as  Stefan  supposed,  it  is  only  strictly  applicable 
to  a  perfectly  black  body.  The  Stefan-Boltzmann  law  may  be 
stated  as  follows  :  —  The  total  radiation  from  a  perfectly  black  body 
is  directly  proportional  to  the  fourth  power  of  the  absolute  tempera- 
ture. If  the  total  radiation  be  denoted  by  S,  we  may  write, 

S  =  CT\  (2) 


where  C  is  a  constant.  If  the  radiation  from  the  sun  be  con- 
sidered solely  as  a  temperature  effect,  its  temperature  may  be 
calculated  by  the  above  equation  expressing  the  Stefan-Boltz- 
mann law.  Employing  available  bolometric  data,  the  tempera- 
ture of  the  sun  may  thus  be  shown  to  be  6200°  absolute. 

The  Displacement  Law  of  Wien.  Having  considered  the  total 
energy  radiated  by  a  given  source,  we  now  come  to  the  considera- 
tion of  the  distribution  of  energy  throughout  the  entire  spectrum 
in  its  relation  to  temperature.  The  results  of  the  experiments 
of  Lummer  and  Pringsheim  |  on  the  distribution  of  energy  in  the 
normal  spectrum  of  a  black  body  are  represented  graphically  in  Fig. 
129.  The  values  of  the  energy  radiated  by  the  source  are  plotted 
as  ordinates  against  the  corresponding  values  of  the  wave-length 
as  abscissae.  It  will  be  observed  that  the  energy  corresponding  to 
a  definite  wave-length  increases  with  the  temperature,  and  that 
each  curve,  or  isothermal,  exhibits  a  distinct  maximum.  The 
position  of  this  maximum  is  displaced  in  the  direction  of  decreas- 
ing wave-length  as  the  temperature  is  raised. 

In  1893,  Wien  §  discovered  the  law  governing  this  displacement 
of  the  energy  maximum  with  temperature.  If  Xmax.  denotes  the 
wave-length  corresponding  to  the  energy  maximum,  and  T  is  the 

*  Sitz.  Ber.  Wiener  Akad.,  79  (II),  391  (1879). 

t  Wied.  Ann.,  22,  291  (1884). 

t  Verh.  deutsch.  phys.  Ges.,  i,  230  (1889);  3,  36  (1901). 

§  Wied.  Ann.,  58,  662  (1896). 


528 


THEORETICAL  CHEMISTRY 


absolute  temperature  of  the  radiating  black  body,  Wien  showed 
that 

Xmax.  X  T  =  constant. 

Furthermore,  Wien  found  that 

Smax.  =  BT\  (3) 

where  Smax.  is  the  maximum  emissivity  corresponding  to  Xmax. 
In  general,  the  emissivity,  S,  is  the  amount  of  energy  radiated  per 


1640" 


Wave-lengthr 
Fig.   129 

second  from  a  narrow  strip  of  the  spectrum  corresponding  to  the 
mean  wave-length,  X.  It  should  be  mentioned,  that  while  the 
experimental  realization  of  a  perfectly  black  body  is  impossible, 
a  very  close  approximation  can  be  obtained  by  the  employment  of 
a  hollow  blackened  sphere,  perforated  by  a  small  hole  to  permit 


PHOTOCHEMISTRY  529 

the  passage  of  the  radiation.  When  this  sphere  is  heated,  prac- 
tically all  of  the  radiation  is  absorbed  by  multiple  reflections  at  the 
inner  surface.  It  has  been  found,  that  where  such  a  black  body  is 
not  available,  or  where  extreme  accuracy  is  not  required,  a  thin 
strip  of  platinum  foil  coated  with  ferric  oxide,  and  heated  electric- 
ally, proves  a  satisfactory  substitute.  It  is  hardly  necessary  to  call 
attention  to  the  fact,  that  the  term  black  body,  as  here  used,  does 
not  imply  a  total  absence  of  color.  At  high  temperatures,  a 
"  black  "  source  of  radiation  may  be  red,  or  even  white.  To  avoid 
confusion,  it  has  been  proposed  to  substitute  the  term  full  radia- 
tor for  the  older  term,  black  body. 

Distribution  of  Energy  throughout  the  Spectrum.  The  experi- 
mental determination  of  the  radiant  energy  corresponding  to  a 
given  wave-length,  really  resolves  itself  into  the  measurement  of 
the  energy  emitted  between  two  contiguous  wave-lengths,  X  and 
X  +  d\.  In  other  words,  we  actually  measure  the  energy  of  a 
very  small  portion  of  the  spectrum  included  between  two  wave- 
lengths which  lie  very  close  together. 

Several  different  formulas  have  been  proposed  for  the  calcula- 
tion of  the  distribution  of  energy  throughout  the  spectrum.  Of 
these,  the  formulas  of  Rayleigh  and  Wien  have  been  found  to 
reproduce  experimental  values  with  considerable  accuracy.  The 
formula  of  Rayleigh  has  the  following  form, 

CT      -- 

.e.  =  —  .  e   xy  •  (4\ 

^       X4  W 

while  that  of  Wien  may  be  written  thus, 

C' 


In  these  two  formulas,  C  and  c'  are  constants,  while  the  other 
symbols  have  their  usual  significance.  The  formula  of  Rayleigh 
has  been  found  to  hold  better  in  the  region  of  the  longer  wave- 
lengths, while  the  reverse  is  true  of  the  formula  of  Wien.  It  should 
be  remembered,  that  both  of  these  formulas  apply  only  to  bodies 
emitting  continuous  spectra. 
High  Temperature  Thermometry.*  Various  optical  methods 

*  For  a  detailed  account  of  optical  pyrometers,  the  student  is  referred  to 
"  High  Temperature  Measurements  "  by  Le  Chatelier  and  Boudouard  trans- 
lated by  Burgess  (John  Wiley  and  Sons,  Inc.). 


530  THEORETICAL  CHEMISTRY 

for  the  measurement  of  high  temperatures  have  been  developed, 
but  a  detailed  treatment  of  these  methods  is  obviously  out  of 
place  in  a  book  of  this  character.  Mention  should  be  made,  how- 
ever, of  two  instruments  which  have  proven  of  great  value  in 
high  temperature  measurements. 

The  optical  pyrometer  of  Fery  is  based  upon  Stefan's  law  of 
total  radiation.  It  consists  of  a  telescope  fitted  with  an  objective 
of  fluorite,  at  the  focus  of  which  is  placed  a  sensitive  thermo- 
couple. In  order  to  determine  the  temperature  of  a  source  of 
radiant  energy,  such  as  a  crucible  of  molten  metal,  the  telescope 
is  directed  toward  the  contents  of  the  crucible  and  the  image  is 
focussed  on  the  thermo- junction  by  means  of  an  adjustable  eye- 
piece. The  resulting  electric  current  is  then  measured  by  means 
of  a  galvanometer. 

In  the  optical  pyrometer  of  Holborn  and  Kurlbaum,  use  is  made 
of  luminous  radiations  only.  In  this  instrument,  the  current 
through  a  small  incandescent  lamp  is  varied  until  its  light  is  just 
eclipsed  by  that  from  the  hot  body.  When  the  point  of  balance 
has  been  reached,  the  incandescent  filament  and  the  hot  body 
have  the  same  temperature.  The  pyrometer  is  calibrated  by  de- 
termining the  current  necessary  to  raise  the  filament  of  the  lamp 
to  the  temperature  of  the  standard  black  body,  the  temperature 
of  the  latter  being  determined  by  means  of  a  thermocouple.  Of 
course,  when  the  instrument  is  used  to  determine  the  temperature 
of  sources  of  radiant  energy  which  differ  widely  in  character  from 
that  of  a  perfectly  black  body,  the  accuracy  of  the  measure- 
ments is  lessened,  but  even  in  an  extreme  case,  such  as  that  pre- 
sented by  polished  platinum  at  950°,  the  error  does  not  exceed 
74°.  The  importance  of  optical  pyrometers  in  photochemical 
investigations  lies  chiefly  in  the  determination  of  energy  curves 
of  light  sources,  and  in  the  absolute  measurement  of  radiant 
energy. 

Luminescence.  As  has  already  been  pointed  out,  it  is  cus- 
tomary to  distinguish  between  pure  temperature  radiation  and 
luminescence.  The  latter  term  is  applied  to  all  cases  where  chemi- 
cal or  electrical  energy  is  transformed  directly  into  radiant  energy. 
The  various  types  of  luminescence  may  be  conveniently  classified 
in  the  following  manner: 


PHOTOCHEMISTRY  531 

Type  of  Luminescence.  Origin  of  Radiation. 

(1)  Photo-luminescence.  Preliminary  exposure  of  the  lumi- 

(a)  Fluorescence,  nous    substance    to    some    external 

(&)  Phosphorescence.  source  of  radiant  energy. 

(2)  Thermo-luminescence.  Stimulation  by  heat,  but  at  a  tem- 

perature considerably  lower  than  that 
required  for  pure  temperature  radia- 
tion. 

(3)  Chemi-luminescence.  Chemical  reaction. 

(4)  Tribo-luminescence.  Fracture  or  cleavage  of  crystals. 

(5)  Cathodo-luminescence.  Electric  discharge. 

(6)  Radio-luminescence.  Radioactivity. 

By  the  term  fluorescence  is  meant  the  phenomenon  of  the  emission, 
by  an  illuminated  medium,  of  light  of  a  different  wave-length  from 
that  of  the  incident  radiation.  In  general,  the  wave-length  of  the 
transformed  radiation  is  greater  than  that  of  the  incident  radia- 
tion. This  law,  to  which  numerous  exceptions  have  been  dis- 
covered, was  first  enunciated  by  Stokes.  When  the  incident 
radiation  is  cut  off,  fluorescence  ceases. 

Among  the  numerous  substances  which  are  known  to  exhibit  the 
phenomenon  of  fluorescence,  may  be  mentioned  fluorite  (from 
which  the  phenomenon  derived  its  name),  uranium  glass,  petro- 
leum, solutions  of  organic  dyestuffs,  and  quinine  sulphate.  The 
vapors  of  sodium,  mercury,  and  iodine  have  recently  been  found 
by  Wood  to  fluoresce  brilliantly. 

There  are  also  many  substances  known  which  continue  to 
emit  light  for  some  time  after  the  external  light-stimulus  is 
removed.  This  phenomenon  is  termed  phosphorescence,  and 
appears  to  be  governed  by  Stokes  law  for  fluorescence.  The 
property  of  phosphorescence  is  apparently  limited  to  anhydrous 
substances. 

The  sulphides  of  the  alkaline  earths  may  be  mentioned  as  ex- 
amples of  phosphorescent  substances.  The  investigations  of 
Lenard  and  Urbain  have  revealed  the  interesting  fact  that  the 
presence  of  a  trace  of  one  of  the  heavy  metals  greatly  intensifies 
the  light  emitted  by  a  phosphorescent  substance. 

The  phenomenon  of  thermo-luminescence  calls  for  little  com- 
ment. There  seems  to  be  an  intimate  connection  between  thermo- 
luminescence  and  phosphorescence,  since  the  substances  exhibit- 
ing the  former  phenomenon  must  be  exposed  initially  to  light, 
otherwise  they  do  not  emit  any  visible  radiation  on  gentle  heating. 


532  THEORETICAL  CHEMISTRY 

Chemi-luminescence  is  a  phenomenon  which  accompanies  many 
chemical  reactions.  Thus,  the  precipitation  of  sodium  chloride 
from  its  saturated  solution  by  hydrochloric  acid  gas,  is  accom- 
panied by  an  emission  of  light,  which  may  readily  be  seen  if  the 
reaction  is  carried  out  in  a  dark  room. 

When  certain  crystals,  such  as  those  of  cane  sugar,  are  either 
crushed,  or  simply  rubbed  together,  flashes  of  light  are  emitted. 
This  phenomenon  is  known  as  tribo-luminescence. 

Cathodo-  and  radio-luminescence  may  be  considered  as  sub- 
divisions of  the  more  inclusive  term,  electro-luminescence.  At- 
tention has  already  been  called  to  the  fact,  that  the  residual  gas  in 
a  vacuum  tube  is  rendered  luminous  by  the  passage  of  the  electric 
discharge,  and  also  that  certain  minerals  become  phosphorescent 
when  placed  in  the  path  of  the  cathode  rays.  These  may  be  taken 
as  examples  of  cathodo-luminescence.  The  luminosity xof  a  screen 
coated  with  crystals  of  zinc  sulphide,  when  subjected  to  the  ac- 
tion of  the  a-particles  shot  out  from  a  radio-active  substance,  has 
also  been  mentioned  in  a  previous  chapter.  This  is  clearly  an 
instance  of  radio-luminescence. 

Having  briefly  reviewed  the  different  processes  involved  in  the 
production  of  light  we  now  turn  to  a  consideration  of  the  chemical 
phenomena  resulting  from  exposure  to  light. 

Photochemical  Action.  The  development  of  the  green  color 
of  plants  under  the  influence  of  the  rays  of  the  sun,  and  the  reverse 
process  of  bleaching  in  darkness,  were  probably  the  first  photo- 
chemical reactions  to  be  observed.  To-day  it  is  known,  that  light 
has  the  power  of  initiating,  or  accelerating,  every  variety  of  chem- 
ical change.  This  statement  may  be  illustrated  by  the  following 
typical  photochemical  reactions :  —  The  polymerization  of  an- 
thracene, the  depolymerization  of  ozone,  the  transformation  of 
malei'noid  into  fumaroid  forms,  the  hydrolysis  of  acetone,  the 
oxidation  of  lead  sulphide,  and  the  reduction  of  silver  salts.  That 
such  a  variety  of  photochemical  reactions  should  result  from  ex- 
posure to  mixed,  or  heterogeneous,  light  is  due  to  the  selective 
absorption  of  each  particular  chemical  system. 

Photochemical  action  is  not  limited  to  the  waves  of  the  visible 

spectrum  alone,  but  extends  from  the  red  end  of  the  spectrum 

(wave-length  800  WJL)  *  into  the  ultra-violet  region  (wave-length 

200  ju/x).     In  fact,  the  shorter  wave-lengths  of  the  ultra-violet 

*  1  MM-7  =  10-7  cm. 


PHOTOCHEMISTRY  533 

region  of  the  spectrum  have  been  found  to  be  the  most  active 
photochemically. 

Laws  of  Grotthuss.  The  two  fundamental  generalizations 
of  photochemistry  were  first  enunciated  by  Grotthuss,  in  1818. 
These  generalizations  may  be  stated  as  follows: 

(1)  Only  those  rays  of  light  which  are  absorbed,  produce  chemical 
action. 

(2)  The  action  of  a  ray  of  light  is  analogous  to  that  of  a  voltaic  cell. 
It  has  been  shown  by  Bancroft,*  that  the  second  of  these  two 

laws  is  inadequate  to  account  for  all  of  the  known  facts,  and  there- 
fore, he  proposes  the  following  modification:  —  All  of  the 
radiations  which  are  absorbed  by  a  substance  tend  to  eliminate 
that  substance.  It  is  merely  a  question  of  chemistry  whether  any  re- 
action occurs,  and  what  the  products  of  the  reaction  will  be. 

Quantitative  Relations  Concerning  the  Absorption  of  Light. 
When  a  ray  of  light  enters  an  absorbing  medium,  only  a  certain 
proportion  of  the  incident  radiation  is  absorbed.  The  intensity 
of  the  light  entering  an  absorbing  medium  is  not  equal  to  that 
which  is  incident  on  the  surface  of  the  medium,  owing  to  the  fact 
that  a  portion  of  the  incident  beam  is  reflected.  It  has  been 
found,  that  if  the  thickness  of  the  medium  be  increased  in  arith- 
metical progression,  the  intensity  of  the  transmitted  light  will 
decrease  in  geometrical  progression.  If  the  intensity  of  the 
light  traversing  a  layer  dl  be  denoted  by  7,  then 

-  dl/dl  =  kl,  (6) 

where  A:  is  a  constant,  depending  upon  the  nature  of  the  absorbing 
medium,  and  the  wave-length  of  the  light.  This  constant,  k,  is 
known  as  the  absorption  index.  If  the  initial  intensity  of  the  light 
is  /o,  and  the  total  thickness  of  the  medium  is  d,  equation  (6)  be- 
comes, on  integrating, 

/  =  70  •  e-™.  (7) 

Or,  equation  (7)  may  be  written  in  the  form, 


If  we  replace  e~*  by  a,  then  equation  (7)  becomes, 

7/70  =  <*d.  (9) 

*  Jour.  Phys.  Chem.,  12,  209,  318,  417  (1908);    13,  1,  181,  269,  449,  538 
(1909);   14,  292  (1910). 


534  THEORETICAL  CHEMISTRY 

The  constant,  a,  is  called  the  transparency,  or  the  transmission  co- 
efficient. Bunsen  and  Roscoe  introduced  the  term,  extinction  co- 
efficient. This  quantity  may  be  denned,  as  the  reciprocal  of  that 
thickness  of  the  medium  which  reduces  the  intensity  of  the  trans- 
mitted light  to  one-tenth  of  its  initial  value.  If  the  extinction 
coefficient  be  denoted  by  e,  its  value  may  be  calculated  from  equa- 
tion (7)  in  the  following  manner: 

/  =  Jo  •  10-*,  (10) 


e=loge.  (11) 

It  is  evident,  that  e~*  is  identical  with  10~*,  or  e  =  mk,  where  m 
represents  the  modulus  of  the  Naperian  system  of  logarithms. 

In  1852,  Beer  *  enunciated  an  important  law  concerning  the  in- 
fluence of  concentration  on  absorption.  Beer's  law  may  be  stated 
as  follows  :  —  The  absorption  of  light  by  different  concentrations  of 
the  same  solute  dissolved  in  the  same  solvent  is  an  exponential  func- 
tion of  the  concentration,  provided  the  thickness  of  the  absorbing 
medium  be  maintained  constant.  It  follows  from  this  law,  that 

/  =  Jo**,  (12) 


where  c  is  the  concentration  of  the  solution. 

If  Beer's  law  is  valid,  the  ratio,  c/e  =  A,  known  as  the  absorp- 
tion ratio,  should  be  constant.  Beer's  law  has  been  tested  with  a 
large  number  of  solutions  and  has  been  found  to  hold  only  in  those 
cases  where  no  change  in  the  solute  occurs  when  the  concentration 
of  the  solution  is  altered. 

Photochemical  Extinction.  According  to  the  first  law  of 
Grotthuss,  only  the  absorbed  light  is  chemically  active.  The 
converse  of  this  law,  viz.,  that  every  substance  which  absorbs 
light  undergoes  chemical  change,  apparently  does  not  hold.  Fur- 
thermore, only  a  portion  of  the  rays  absorbed  by  a  light-sensi- 
tive substance  are  directly  involved  in  effecting  chemical  change. 
For  example,  while  an  alkaline  copper  tartrate  solution  shows 
marked  absorption  in  the  infra-red,  red,  yellow,  and  ultra-violet, 
it  has  been  proven,  that  the  photochemical  reduction  of  the  copper 
salt  to  cuprous  oxide  is  due  to  the  action  of  the  ultra-violet  rays 
alone. 

The  first  quantitative  measurements  of  the  absorption  of  light 

*  Pogg.  Ann.,  86,  78  (1852), 


PHOTOCHEMISTRY  535 

by  a  reacting  system  were  made  by  Bunsen  and  Roscoe.*  These 
investigators  found,  that  the  absorption  of  light  by  hydrogen  and 
chlorine,  taken  separately,  was  less  than  the  absorption  of  the  re- 
acting mixture  of  the  two  gases.  From  this  result,  they  concluded 
that  in  a  photochemical  reaction,  the  absorption  of  light  by  the  re- 
acting system  is  greater  than  the  sum  of  the  individual  absorp- 
tions of  the  reacting  substances.  The  absorption  of  light  by  the 
reacting  system,  over  and  above  the  ordinary,  or  "  optical"  ab- 
sorption of  the  reacting  substances,  they  called  photochemical 
extinction.  While  the  phenomenon  of  photochemical  extinction 
is  regarded  by  many  physical  chemists  as  having  doubtful  signi- 
ficance, nevertheless  the  distinction  between  purely  optical  ab- 
sorption on  the  one  hand,  and  chemical  absorption  on  the  other, 
is  quite  generally  accepted. 

If  the  distinction  drawn  by  Bunsen  and  Roscoe  between  optical 
and  chemical  absorption  be  accepted,  then  all  photochemical  re- 
actions may  be  regarded  as  belonging  to  one,  or  the  other,  of  two 
classes,  as  follows:  —  (1)  reactions  in  which  light  does  work 
against  chemical  affinity,  the  work  being  equivalent  to  the  photo- 
chemical extinction;  or  (2)  reactions  in  which  the  light  functions 
merely  as  a  catalyst.  In  reactions  belonging  to  the  first  class,  the 
light  is  considered  to  be  the  agent  which  actually  initiates  chemi- 
cal change,  whereas  in  reactions  of  the  second  class,  the  light  is 
assumed  to  accelerate  reactions  which  would  otherwise  proceed 
at  a  slower  rate. 

Kinetics  of  Photochemical  Reactions.  Let  A  and  B  repre- 
sent two  chemically  distinct  substances,  and  let  us  assume  that  the 
reaction, 


takes  place  under  the  influence  of  light.  The  course  of  the  re- 
action can  be  followed  in  the  usual  manner,  by  determining  the 
amount  of  either  constituent  which  is  present  at  any  definite 
time.  There  are  certain  factors,  however,  which  render  such  kin- 
etic measurements  more  difficult  in  the  case  of  a  photochemical 
reaction  than  in  that  of  an  ordinary  chemical  reaction.  Let  us 
assume  that  the  above  reaction  is  homogeneous  and  of  the  first 
order,  and  also  that  it  takes  place  in  homogeneous  solution.  It  is 
apparent  that  if  the  system  is  illuminated  from  one  side,  as  shown 

*  OstwalcTs  Klassiker,  No.  38. 


536 


THEORETICAL  CHEMISTRY 


in  Fig.  130,  the  rate  at  which  A  undergoes  transformation  into 
B  will  depend  upon  the  thickness,  d,  of  the  absorbing  layer.  Hence, 
if  the  velocity  of  the  transformation  be  denoted  by  dB/dt,  we 
may  write, 

dB/dt  =  k[A],  (13) 

in  which  the  value  of  the  constant,  k,  will  not  only  be  a  function  of 
the  intensity  and  wave-length  of  the  light,  but  also  of  position. 

Furthermore,  the  variation  in 
the  velocity  of  the  reaction  in 
successive  layers  will  cause 
differences  in  concentration  which 
will  tend  to  become  equalized 
by  the  process  of  diffusion. 
From  this  example,  it  is  obvious 
that  the  extent  to  which  light 
is  absorbed  is  dependent  upon 


Fig.  130 


the  dimensions  of  the  absorbing  system. 

If  light  be  regarded  as  a  material  substance,  then  we  may  con- 
veniently consider  its  absorption  as  analogous  to  the  diffusion  of 
a  gas  into  a  liquid,  and  the  light  intensity  at  any  point  may  be 
regarded  as  the  analogue  of  concentration,  or  active  mass.  Ac- 
cording to  the  law  of  mass  action,  the  velocity  of  a  chemical 
reaction  is  expressed  by  the  equation, 

dx/dt  =  kcini  -  (tf1*  •  •  -  —  k'c\n*  •  Czn2'  •  •  -  , 

where  ci,  C2,  etc.,  are  the  concentrations  of  the  substances  entering 
into  the  reaction,  and  HI,  n2,  etc.,  are  the  coefficients  derived  from 
the  chemical  equation,  and  indicating  the  order  of  the  reaction. 
In  applying  this  equation  to  photochemical  reactions,  the  follow- 
ing possibilities  must  be  borne  in  mind : 

(1)  The  reaction  may  take  place  in  successive  stages.     Under 
these  conditions  the  experimentally  measured  velocity  will  corre- 
spond to  that  of  the  slowest  reaction. 

(2)  Side  reactions  may  occur,  with  the  formation  of  products 
quite  different  from  those  resulting  from  the  main  reaction. 

(3)  There  may  be  catalysis.     In  fact,  this  phenomenon  is  fre- 
quently met   with   in    the    study    of    photochemical    reactions. 
Nernst  has   pointed   out  that  the  velocity  constants,  k  and  "A/, 
in  the  foregoing  equation,   may  conveniently  be  considered  as 
being  directly  proportional  to  the  intensity  of  the  light,  for  light 


PHOTOCHEMISTRY  537 

of  the  same  wave  length.  Owing  to  the  absorption  of  the  me- 
dium, this  intensity  will  be  a  function  of  position  in  the  medium. 
In  applying  the  law  of  mass  action  to  photochemical  reactions, 
it  is  important  to  note  that,  as  a  general  rule,  the  exponents  n\t 
HZ,  etc.,  in  the  equation  of  the  so-called  dark-reaction,  are  not  iden- 
tical with  the  exponents  vi,  *>2,  etc.,  of  the  light-reaction.  In  other 
words,  the  order  of  a  chemical  reaction  is  usually  different  in  the 
light  from  what  it  is  in  the  dark.  The  photochemical  exponents 
vi,  vz,  etc.,  are  never  greater  than,  and  are  seldom  equal  to,  the 
corresponding  exponents  n\,  nz,  etc.,  of  the  dark-reaction. 

Thus,  Bodenstein  *  showed  that  the  dissociation  of  hydriodic 
acid,  in  the  dark,  is  a  reaction  of  the  second  order  and  may  be  rep- 
resented by  the  equation, 

2  HI  <=>  H2  +  I,, 

whereas,  when  hydriodic  acid  dissociates  in  the  light,  the  reaction 
is  of  the  first  order,  as  shown  by  the  equation, 

HI  <=>  H  +  I. 

In  this  case,  the  light  merely  accelerates  the  velocity  of  the  dark- 
reaction. 

The  action  of  light  on  the  reverse  reaction  may  be  such  as  to 
oppose  its  usual  course  in  the  dark.  In  this  case,  the  resulting 
photochemical  equilibrium,  or  so-called  photo-stationary  state,  will 
not  be  identical  with  the  corresponding  chemical  equilibrium.  It 
should  be  observed,  that  the  photo-stationary  state  differs  from 
ordinary  chemical  equilibrium,  in  that  its  permanency  is  wholly 
dependent  upon  the  constancy  of  the  source  of  illumination;  i.e., 
when  the  light  is  cut  off,  the  photo-stationary  state  shifts  to  the 
ordinary  chemical  equilibrium,  provided  the  reaction  is  reversible. 
It  has  also  been  found,  that  the  temperature  coefficients  of  most 
photochemical  reactions  are  negligible. 

Becquerel  f  discovered  that  silver  chloride,  which  had  been 
precipitated  in  the  dark,  was  only  sensitive  to  short  wave-lengths 
of  light,  whereas  silver  chloride,  which  had  been  exposed  for  a  few 
moments  to  sunlight,  became  sensitive  to  all  wave-lengths  in  the 
visible  spectrum,  and  also  to  the  shorter  wave-lengths  of  the  infra- 
red. This  phenomenon,  which  has  been  observed  with  various  sub- 

*  Zeit.  phys.  Chem.,  22,  23  (1897). 
t  Ann.  Chim.  Phys.  [3],  9,  257  (1843). 


538  THEORETICAL   CHEMISTRY 

stances,  was  first  studied  systematically  by  Bunsen  and  Roscoe,* 
who  termed  it  photochemical  induction.  Employing  a  mixture  of 
hydrogen  and  chlorine  gases,  they  found  that  under  constant 
illumination,  the  velocity  of  formation  of  hydrochloric  acid, 
which  was  hardly  appreciable  at  first,  increased  rapidly  to  a 
maximum,  and  then  remained  constant.  The  interval  of  time 
required  for  the  reaction  to  attain  its  maximum  velocity  is  known 
as  the  period  of  induction.  It  has  been  found,  that  in  almost 
every  photochemical  reaction  there  is  a  similar  period  of  initial 
perturbation.  The  phenomenon  has  been  thoroughly  investi- 
gated by  Burgess  and  Chapman, f  who  arrived  at  the  conclusion, 
that  induction  effects  are  to  be  ascribed  to  the  presence  of  minute 
traces  of  various  impurities,  such  as  gases  or  water  vapor,  ad- 
sorbed by  the  walls  of  the  reaction-vessel.  We  may  therefore 
conclude,  that  induction  effects  are  not  characteristic  of  photo- 
chemical reactions. 

Classification  of  Photochemical  Reactions.  According  to 
Sheppard  ("  Photochemistry  ")  photochemical  reactions  may  be 
conveniently  classified  in  the  following  manner: 

(1)  Reversible  reactions,  i.e.,  reactions  in  which  the  products 
formed  under  the  influence  of  light  react  to  reproduce  the  original 
system  when  the  light  is  removed. 

(2)  Irreversible  reactions,  i.e.,  reactions  in  which  the  light  pro- 
motes   transformation    to    a   more    stable    system.     Irreversible 
reactions  are  subdivided  into  — 

(a)  Complete  reactions;  and 

(b)  Pseudo-reversible  reactions. 

The  polymerization  of  anthracene  may  be  taken  as  an  example 
of  a  reversible  photochemical  reaction.  This  reaction  may  be 
represented  as  taking  place  according  to  the  equation, 

light 

2  Cl4Hio  <=^  C2sH2o, 
dark 

the  upper  arrow  indicating  the  direction  of  the  light-reaction 
(polymerization),  and  the  lower  arrow  that  of  the  dark-reaction 
(de-polymerization) . 

An  illustration  of  a  completely  irreversible  photochemical  re- 
action is  afforded  by  a  mixture  of  hydrogen  and  chlorine  gases 

*  Fogg.  Ann.,  100,  481  (1857). 

f  Jour.  Chem.  Soc.,  89,  1402  (1906). 


PHOTOCHEMISTRY  539 

which  combine,  on  exposure  to  light,  to  form  hydrochloric  acid 
gas,  according  to  the  equation, 

light 
H2  +  C12  -*  2  HC1. 

If  the  hydrochloric  acid  is  removed  by  solution  in  water  as  fast 
as  it  is  formed,  the  velocity  of  the  reaction  will  be  directly  pro- 
portional to  the  intensity  of  the  light.  Bunsen  and  Roscoe  * 
made  use  of  the  foregoing  facts  in  the  construction  of  their 
actinometer. 

The  reduction  of  ferric  oxalate  may  be  taken  as  an  example  of 
a  pseudo-reversible  photochemical  reaction.  This  substance  is 
reduced  to  ferrous  oxalate  on  exposure  to  light  as  shown  by  the 
equation, 

light 

Fe2(C204)3  ->  2  Fe  (C204)  +  C02. 

In  the  dark,  ferrous  oxalate  is  re-oxidized  to  ferric  oxalate  by  the 
oxygen  of  the  air.  While  the  initial  substance  is  reproduced,  it  is 
evident  that  the  reaction  is  not  strictly  reversible. 

Actinometers.  A  number  of  different  forms  of  apparatus  have 
been  devised  for  measuring  the  chemical  action  of  light;  such  in- 
struments are  known  as  actinometers. 

The  hydrogen-chlorine  actinometer  is  based  upon  the  well- 
known  fact,  that  the  speed  of  the  reaction  between  hydrogen  and 
chlorine  varies  greatly  with  the  intensity  of  illumination.  Bun- 
sen  and  Roscoe,  f  guided  by  the  experiments  of  Draper,  con- 
structed an  actinometer  in  which  the  rate  of  combination  of  hydro- 
gen and  chlorine  could  be  measured,  by  allowing  the  hydrochloric 
acid  formed  to  dissolve  in  water,  and  noting  the  resulting  diminu- 
tion of  volume.  A  diagram  of  this  apparatus  is  given  in  Fig.  131. 


131 


The  apparatus  is  filled  with  a  mixture  of  equal  parts  of  hydrogen 
and  chlorine,  obtained  by  the  electrolysis  of  a  solution  of  hydro- 
chloric acid.  The  bulb  A,  containing  water,  is  connected  at  one 
end  with«a  tube  fitted  with  a  stop-cock  5,  and  at  the  other  end 

*  "  Photochemische  Untersuchungen,"  Ostwald's  Klassiker,  No.  34. 
f  Pogg.  Ann.,  100,  43  (1857);   101,  235  (1857). 


540  THEORETICAL  CHEMISTRY 

with  a  horizontal  tube  terminating  in  a  reservoir  D,  which  also 
contains  water.  When  the  water  has  become  saturated  with  the 
constituents  of  the  gaseous  mixture,  B  is  closed  and  the  entire 
apparatus  is  protected  from  light.  When  it  is  desired  to  measure 
the  photochemical  action  of  a  source  of  light,  the  bulb  A  is  un- 
covered and  the  light  is  allowed  to  fall  upon  it.  Some  of  the  hydro- 
gen and  chlorine  will  combine,  and  the  hydrochloric  acid  formed 
will  be  absorbed  by  the  water  in  A]  the  column  of  water  in  the 
horizontal  tube  will  move  to  the  right,  the  magnitude  of  the  move- 
ment being  measured  on  the  scale  C.  In  this  manner,  the  amount  of 
photochemical  action  can  be  determined.  An  objection  to  the  use 
of  hydrogen  and  chlorine  in  the  actinometer  is  the  danger  of  vio- 
lent explosions  when  the  illumination  is  too  intense.  To  remove 
this  objection,  Burnett  replaced  the  hydrogen  of  the  mixture  by 
carbon  monoxide. 

Action  of  Light  on  the  Silver  Halides.  Probably  the  most 
familiar,  and  undoubtedly  one  of  the  most  important,  photo- 
chemical reactions  is  that  which  takes  place  on  the  exposure  of  a 
photographic  plate.*  Luther  f  has  shown  that  when  a  pure  sil- 
ver halide  is  exposed  to  light,  it  undergoes  reduction  according  to 
the  equation, 

light 

2  AgX  -*  Ag2X  +  X, 

where  X  may  be  chlorine,  bromine,  or  iodine.-  On  removing  the 
light,  the  sub-halide  recombines  with  the  free  halogen  as  shown 
by  the  equation, 

dark 

Ag2X  +  X  ->  2  AgX. 

In  other  words,  the  reaction  is  strictly  reversible  and  a  well-de- 
fined photo-stationary  state  results  from  a  given  intensity  of 
illumination.  The  investigations  of  BakerJ  make  it  appear  quite 
probable,  that  the  photochemical  reduction  of  the  silver  halides  is 
dependent  upon  the  presence  of  a  minute  trace  of  water  vapor  as  a 
catalyst.  In  the  photographic  plate,  the  silver  halide  is  embedded 
in  gelatine,  which  not  only  accelerates  the  rate  of  reduction  of  the 

*  For  an  excellent  treatment  of  the  chemistry  of  photography  the  student 
is  recommended  to  consult  "  Photography  for  Students  of  Physics  and  Chem- 
istry," by  Louis  Derr  (Macmillan);  Or  "  Photochemie  und  Beschreibung  der 
photographischen  Chemikalien,"  by  H.  W.  Vogel. 

t  Zeit.  phys.  Chem.,  30,  628  (1899). 

$  Jour.  Chem.  Soc.,  61,  782  (1892). 


PHOTOCHEMISTRY  541 

silver  halide,  but  also  causes  the  reaction  to  become  irreversible. 
The  alteration  in  the  behavior  of  the  silver  halide,  brought  about 
by  the  presence  of  gelatine,  is  due  to  the  fact  that  the  latter 
reacts  with  the  free  halogen,  according  to  the  equation, 

light 

2  AgBr  +  gelatine  — »  Ag2Br  +  brominated  gelatine. 

The  continuous  removal  of  the  liberated  bromine  by  the  gelatine 
is  an  example  of  what  is  known  as  photochemical  sensitization. 
Another  example  of  photochemical  sensitization  is  afforded  by  a 
mixture  of  benzene  and  silver  chloride.  The  normal  darkening 
of  the  silver  salt  is  markedly  increased  by  the  presence  of  the 
benzene,  which  combines  with  the  chlorine  as  rapidly  as  it  is  set 
free  by  the  action  of  the  light  on  the  silver  halide. 

While  the  silver  bromide  of  the  photographic  plate  is  extremely 
sensitive  to  the  shorter  wave-lengths,  corresponding  to  the  violet 
and  ultra-violet  regions  of  the  spectrum,  it  is  only  reduced  by  the 
longer  wave-lengths  after  prolonged  exposure.  It  has  been  found, 
however,  that  the  addition  of  certain  dyestuffs,  such  as  eosine  and 
Congo-red,  renders  the  silver  halide  sensitive  to  the  longer  wave- 
lengths of  the  spectrum.  This  phenomenon,  which  is  known  as 
optical  sensitization,  must  be  carefully  distinguished  from  chemical 
sensitization,  to  which  reference  has  already  been  made.  It  is  to 
be  noted  that  an  optical  sensitizer  does  not  absorb,  chemically, 
any  of  the  products  of  the  reaction.  Photographic  plates  which 
have  been  sensitized  in  this  way  are  commonly  known  as  ortho- 
chromatic  plates. 

All  of  the  dyestuffs  which  can  function  as  optical  sensitizers 
have  been  found  to  exhibit  anomalous  refraction;  i.e.,  for  wave- 
lengths slightly  longer  than  those  absorbed,  these  substances 
possess  an  abnormally  large  refractive  index,  in  consequence  of 
which  the  refracted  waves  exert  the  same  effect  upon  the  silver 
halide  as  the  shorter  wave-lengths  of  the  spectrum.  Although  it 
is  n'ot  an  essential  property,  it  is  generally  found  that  optical  sensi- 
tizers. are  fluorescent.  As  an  example  of  optical  sensitization, 
where  the  sensitizer  is  non-fluorescent,  we  may  take  the  photo- 
chemical reduction  of  mercuric  chloride  in  the  presence  of  am- 
monium oxalate.  This  reaction  takes  place  according  to  the 
equation, 

light 

2  HgCl2  +  (NH4)2C204  -» Hg2Cl2  +  NH4C1  +  2  CO2. 


542  THEORETICAL  CHEMISTRY 

The  presence  of  the  non-fluorescent  ferric  ion,  Fe'",  has  been 
shown  by  Winther  *  to  be  an  effective  optical  sensitizer  in  the  re- 
action. The  ferric  ion  is  reduced  to  the  ferrous  state,  while  the 
oxalic  acid  undergoes  oxidation.  It  was  pointed  out  by  Eder  f 
that  this  reaction  is  well  adapted  for  actinometric  measurements, 
since  the  amount  of  mercurous  chloride  precipitated  is  directly 
proportional  to  the  intensity  of  the  light. 

Photochemical  After-Effect.  Certain  photochemical  reactions 
are  known  in  which  the  reaction  has  been  found  to  proceed  even 
after  the  light  stimulus  is  removed.  This  phenomenon  is  known 
as  the  photochemical  after-effect.  The  velocity  of  a  reaction  of 
this  kind  is  different  from  that  of  either  the  light-,  or  the  dark- 
reaction.  It  has  also  been  found,  that  if  a  portion  of  the  reac- 
tion-mixture, which  has  already  been  exposed  to  the  light,  be 
added  to  a  fresh  unexposed  portion,  the  latter  immediately  com- 
mences to  decompose.  Thus,  a  solution  of  iodoform  in  chloro- 
form becomes  brown,  on  exposure  to  light,  due  to  liberation  of 
iodine.  If  the  solution  is  removed  from  the  light,  the  decomposition 
will  continue  for  several  days,  while  if  a  small  portion  of  the  par- 
tially decomposed  solution  be  added  to  a  freshly  prepared  solution, 
the  latter  will  commence  to  decompose.  The  photochemical  after- 
effect can  be  readily  explained,  if  we  assume  that  the  action  of  light 
gives  rise  to  heterogeneous  nuclei,  which  persist  for  a  sufficient 
time  after  the  light  is  removed  to  act  as  centers  around  which  the 
reaction  can  proceed  throughout  the  unexposed  portion  of  the 
mixture. 

Assimilation  of  Carbon  Dioxide.  A  photochemical  reaction 
of  special  interest  to  the  biologist  is  that  in  which  atmospheric 
carbon  dioxide  is  taken  up  by  plants,  under  the  influence  of  solar 
radiation.  While  the  reaction  is  exceedingly  complex,  it  may  be 
regarded  as  taking  place  according  to  the  hypothetical  equation, 

light 

6  C02  +  5  H2O  —  C6Hi0O5  +  6  O2. 

(starch) 

This  reaction  represents  a  gain  in  energy  amounting  to  approxi- 
mately 685  calories  per  formula- weight  of  starch.  It  is  quite 
probable,  however,  that  the  initial  product  of  the  reaction  is  for- 
maldehyde, and  that  subsequently  the  latter  substance  undergoes 
polymerization,  with  the  formation  of  starch  and  other  carbo- 

*  Zeit.  wiss.  Phot.,  7,  409  (1909). 
t  Sitz.  her.  Wien.  Akad.  (1879). 


PHOTOCHEMISTRY  543 

hydrates.  The  green  coloring  matter  of  the  leaves  of  plants, 
known  as  chlorophyll,  also  plays  an  important  part  in  the  assimi- 
lation of  carbon  dioxide,  but  beyond  the  fact  that  its  action  is  not 
catalytic,  little  can  be  stated  as  to  the  manner  in  which  it  func- 
tions in  the  reaction.  The  velocity  of  the  reaction  has  been  found 
to  attain  its  maximum  value  in  yellow  and  green  light,  a  result 
which  is  in  complete  agreement  with  the  fundamental  law  of  Grott- 
huss,  that  only  those  rays  which  are  absorbed  are  active  chemi- 
cally. As  has  already  been  stated,  the  temperature  coefficients 
of  photochemical  reactions  are  generally  very  small.  This  is 
not  true,  however,  of  the  reaction  under  consideration.  It  has 
been  found  that  the  rate  at  which  carbon  dioxide  is  assimilated 
by  a  plant  is  nearly  doubled  for  a  rise  in  temperature  of  10°  (see 
p.  376). 

Photochemical  Synthesis.  While  it  has  long  been  known, 
that  light  is  capable  of  effecting  the  synthesis  of  complex  organic 
compounds  in  the  living  plant  from  carbon  dioxide  and  water,  it 
is  only  recently,  that  its  efficiency  in  bringing  about  a  great  variety 
of  organic  reactions  has  been  fully  recognized.  Owing  to  the  re- 
searches of  Ciamician  and  others  in  this  field,  numerous  photo- 
chemical syntheses  of  considerable  practical  value  to  the  organic 
chemist  have  been  discovered.  It  must  suffice  here  to  mention  a 
few  such  typical  photochemical  syntheses. 

Alcohols  may  be  oxidized  in  successive  steps,  the  action  of  the 
light  closely  resembling  the  action  of  ferments.  Methyl  alcohol 
and  acetone  react  to  form  isobutylene  glycol  according  to  the 
equation, 

CH3      light      CH3 

CH3  -  OH  +  CO        -»      COH  -  CH2OH 

CH,  CH3 

The  action  of  light  has  been  found  to  be  especially  favorable  to 
such  processes  as  the  foregoing,  involving  reciprocal  oxidation  and 
reduction.  Ortho-nitro-benzaldehyde,  in  the  presence  of  ethyl 
alcohol,  reacts  to  form  the  ethyl  ester  of  o-nitroso-benzoic  acid,  as 
shown  by  the  equation, 

N02          ,O  light       NO         /O 

CH3  /V 

O.CH2-CH3 
CH2-OH 


544  THEORETICAL  CHEMISTRY 

Photoelectric  Cells.  The  absorption  of  light  by  any  one  of 
the  three  states  of  matter  is  invariably  accompanied  by  a  change 
in  electrical  condition.  The  different  electric  effects  accompany- 
ing the  absorption  of  radiant  energy  have  been  classified  by  Shep- 
pard  in  the  following  manner: 

(1)  lonization  with  a  corresponding  increase  of  conductance  in 
gases,  liquids,  and  solids,  caused  by  transmission  of  light. 

(2)  Indirect  ionization  of  a  gas,  due  to  reflection,  or  emission  of 
electrons  from  the  surface  of  a  contiguous  denser  phase. 

(3)  Development  of  electromotive  forces  in  cells  of  the  following 
type: 


Electrode  A 

(light) 


Conductor, 
Di-electric 


Electrode  A , 


(dark) 


where  the  two  electrodes  are  separated  by  a  medium  which  is  par- 
tially a  conductor  and  partially  a  di-electric. 

The  treatment  of  the  first  and  second  of  these  three  classes  of 
photoelectric  phenomena  properly  lies  within  the  domain  of  pure 
physics.  The  third  class,  however,  includes  a  number  of  photo- 
galvanic  combinations  of  considerable  interest  and  importance  to 
the  physical  chemist. 

The  first  investigation  of  photoelectric  combinations  was  under- 
taken by  E.  Becquerel,*  in  1839.  He  prepared  a  cell  consisting  of 
identical  plates  of  pure  silver,  coated  with  a  silver  halide,  and 
immersed  in  dilute  sulphuric  acid  as  an  electrolyte.  On  exposing 
one  electrode  to  the  light,  while  the  other  was  kept  in  the  dark,  an 
appreciable  electromotive  force  was  developed,  the  current  flowing 
in  the  solution  from  the  darkened  to  the  illuminated  electrode.  If 
a  galvanometer  be  included  in  the  circuit,  the  deflection  of  the 
needle  may  be  taken  as  a  measure  of  the  intensity  of  the  light. 

The  action  of  the  Becquerel  actinometer  has  been  explained  by 
Ostwald  as  follows  :  —  The  incident  light  lessens  the  stability  of 
the  silver  iodide,  which  undergoes  ionization  according  to  the 
equation, 


the  Ag*  ions  giving  up  their  charges  to  the  electrode,  while  the  I' 
ions  enter  the  solution.  For  every  Ag*  ion  which  is  discharged  at 
the  illuminated  electrode,  an  equal  number  of  Ag'  ions  enter  the 

*  Ann.  Chim.  Phys.  [3],  9,  257  (1843). 


PHOTOCHEMISTRY  545 

solution  at  the  darkened  electrode,  thus  charging  the  latter  nega- 
tively. Hence  the  current  flows,  in  the  solution,  from  the  dark- 
ened to  the  illuminated  electrode. 

Rigollot  *  has  constructed  an  electrical  actinometer  in  which  the 
silver  plates  used  by  Becquerel  are  replaced  by  two  oxidized 
copper  electrodes,  while  instead  of  dilute  sulphuric  acid  a  dilute 
solution  of  sodium  chloride  is  used. 

A  number  of  photoelectric  combinations  have  been  investigated 
by  Wildermann,  f  among  which  he  recommends  the  following  as 
being  especially  suitable  for  actinometers : 

(a)  Ag  |  AgBr,  0.1  N  KBr  |  AgBr  |  Ag , 

(light)  (dark) 

Reaction,     2  AgBr  — >  Ag2Br. 

(b)  Cu  |  CuO  |  NaOH  |  CuO  |  Cu , 

(light)  (dark) 

Reaction,    2  CuO  ->  Cu20. 

Of  these  two  cells,  the  latter  is  perhaps  the  better,  since  it  gives  an 
appreciable  electromotive  force  for  light  of  the  same  intensity  and 
varying  wave-length. 

While  all  of  the  theoretical  interpretations  of  the  action  of 
photoelectric  cells  are  more  or  less  inadequate,  the  investigations 
of  Scholl  and  others  make  it  appear  probable  that  negative  elec- 
trons enter  the  electrolyte  from  the  electrode,  while  positive  ions 
are  discharged  on  the  electrode,  thereby  causing  anodic  polariza- 
tion. In  general  the  direction  of  the  photoelectric  current  inside 
the  cell  is  from  the  non-exposed  to  the  exposed  electrode. 

REFERENCE 

Photochemistry.     Sheppard. 

*  Jour,  de  Phys.  [3],  6,  520  (1897). 

t  Plotnikow's  "  Photochemische  Versuchstechnik  "  is  recommended  as  an 
excellent  guide  to  practical  laboratory  methods. 


CHAPTER  XX 
CLASSIFICATION   OF  THE    ELEMENTS 

Early  Attempts  at  Classification.  Many  attempts  were  made 
to  classify  the  elements  according  to  various  properties,  such  as 
their  acidic  or  basic  characteristics  or  their  valence.  In  all  of 
these  systems  the  same  elements  frequently  found  a  place  in  more 
than  one  group,  and  elements  bearing  little  resemblance  to  each 
other  were  classed  together.  The  early  attempts  at  classification, 
based  upon  the  atomic  weights  of  the  elements,  were  not  success- 
ful, owing  to  the  uncertainty  as  to  the  exact  numerical  values  of 
these  constants. 

Prout's  Hypothesis  In  1815,  W.  Prout,  an  English  physician, 
observed  that  the  atomic  weights  of  the  elements,  as  then  given, 
did  not  differ  greatly  from  whole  numbers,  when  hydrogen  was 
taken  as  the  standard.  Hence  he  advanced  the  hypothesis,  that 
the  different  elements  are  polymers  of  hydrogen  As  has  already 
been  pointed  out,  this  hypothesis  led  Stas  to  undertake  his  refined 
determinations  of  the  atomic  weights  of  silver,  lithium,  sodium, 
potassium,  sulphur,  lead,  nitrogen  and  the  halogens.  As  a  result 
of  his  investigations  he  says:  "I  have  arrived  at  the  absolute 
conviction,  the  complete  certainty,  so  far  as  is  possible  for  a  human 
being  to  attain  to  certainty  in  such  matter,  that  the  law  of  Prout 
is  nothing  but  an  illusion,  a  mere  speculation  definitely  contra- 
dicted by  experience."  Notwithstanding  the  fact,  that  Prout's 
hypothesis  as  originally  stated  was  thus  disproved  by  Stas,  it  still 
survived  in  a  modified  form  given  to  it  by  J.  B.  Dumas,  who  sug- 
gested that  one-half  of  the  atomic  weight  of  hydrogen  should  be 
taken  as  the  fundamental  unit  When  Stas  showed  that  his 
experiments  excluded  this  possibility,  Dumas  suggested  that  the 
fundamental  unit  be  taken  as  one-quarter  of  the  atomic  weight 
of  hydrogen.  Having  begun  to  divide  and  subdivide,  there  was  no 
limit  to  the  process,  and  the  hypothesis  fell  into  disfavor,  although 
the  belief  in  a  primal  element,  something  akin  to  the  protyle 
(TTPOJTT;  U\T/)  of  the  ancient  philosophers,  has  survived  and  in  recent 
times  has  reappeared  in  the  modern  theory  of  atomic  structure. 

646 


CLASSIFICATION  OF  THE  ELEMENTS 


547 


Ddbereiner's  Triads.  About  1817,  J.  W.  Dobereiner  *  observed 
that  groups  of  three  elements  could  be  selected  from  the  list  of  the 
elements,  all  of  which  are  chemically  similar,  and  having  atomic 
weights  such  that  the  atomic  weight  of  the  middle  member  is  the 
arithmetical  mean  of  the  first  and  third  members  of  the  group. 
These  groups  of  three  elements  he  termed  triads.  In  the  follow- 
ing table  a  few  of  these  triads  are  given. 

DOBEREINER'S  TRIADS 


Element. 

At.  Wt. 

Mean  atomic 
weight  of 
triads. 

Lithium 

6  94 

) 

Sodium 

23.00 

>     23.02 

Potassium                

39.10 

) 

Calcium       

40.07 

87.63 

>     88.72 

Barium  

137.37 

Chlorine                           .               

35  46 

) 

Bromine                        

79.92 

>     80.69 

Iodine               .    .    

126.92 

Sulphur     

32.07 

Selenium   

79.2 

I     78.78 

Tellurium  

127.5 

Phosphorus                          

31.04 

Arsenic                            

74.96 

i     75.62 

Antimony                  

120.2 

This  simple  relation,  first  pointed  out  by  Dobereiner,  is  clearly 
a  foreshadowing  of  the  periodic  law. 

The  Helix  of  de  Chancourtois.  The  idea  of  arranging  the 
elements  in  the  order  of  their  atomic  weights  with  a  view  to  em- 
phasizing the  relationship  of  their  chemical  and  physical  prop- 
erties, seems  to  have  first  suggested  itself  to  M.  A.  E.  B.  de  Chan- 
courtois f  in  the  year  1862.  On  a  right-circular  cylinder  he  traced 
what  he  termed  a  "  telluric  helix  "  at  a  constant  angle  of  45°  to  the 
axis.  On  this  curve  he  laid  off  lengths  corresponding  to  the 
atomic  weights  of  the  elements,  taking  as  a  unit  of  measure  a 
length  equal  to  one-sixteenth  of  a  complete  revolution  of  the 
cylinder.  He  then  called  attention  to  the  fact,  that  elements  with 

*  Pogg.  Ann.,  15,  301  (1825). 

f  Vis  Tellurique,  Classement  naturel  des  Corps  Simples. 


548 


THEORETICAL   CHEMISTRY 


analogous  properties  fall  on  vertical  lines  parallel  to  the  generatrix. 
Being  a  mathematician  and  a  geologist  he  did  not  express  himself 
in  such  terms  as  would  attract  the  attention  of  chemists  and  con- 
sequently his  work  remained  unnoticed  until  comparatively  recent 
times. 

The  Law  of  Octaves.  In  1864,  J.  A.  R.  Newlands  *  pointed 
out  that  if  the  elements  are  arranged  in  the  order  of  their  atomic 
weights,  the  eighth  element  has  properties  very  similar  to  the 
first;  the  ninth  to  the  second;  the  tenth  to  the  third;  and  so  on, 
or  to  employ  Newlands'  own  words:  "  The  eighth  element  starting 
from  a  given  one  is  a  kind  of  repetition  of  the  first,  like  the  eighth  note 
of  an  octave  in  music."  This  peculiar  relationship,  termed  by 
Newlands  the  law  of  octaves,  is  brought  out  in  the  following  table. 


NEWLANDS  OCTAVES 


H 

Li 

Gl 

B 

C 

N 

O 

F 

Na 

Ms 

Al 

Si 

P 

S 

Cl 

K 

Ca 

Cr 

Ti 

Mn 

Fe 

Notwithstanding  the  fact  that  its  author  was  ridiculed  and  his 
paper  returned  to  him  as  unworthy  of  publication  in  the  proceed- 
ings of  the  Chemical  Society,  this  generalization  must  be  regarded 
as  the  immediate  forerunner  of  the  periodic  law. 

The  Periodic  Law.  Quite  independently  of  each  other,  and 
apparently  in  ignorance  of  the  work  of  Newlands  and  de  Chan- 
courtois,  Mendeleeff  f  in  Russia  and  Lothar  Meyer  in  Germany, 
gained  a  far  deeper  insight  into  the  relations  existing  between 
the  properties  of  the  elements  and  their  atomic  weights.  In  1869, 
Mendeleeff  wrote:  —  "  When  I  arranged  the  elements  according 
to  the  magnitude  of  their  atomic  weights,  beginning  with  the 
smallest,  it  became  evident  that  there  exists  a  kind  of  periodicity 
in  their  properties.  I  designate  by  the  name  l  periodic  law/  the 
mutual  relations  between  the  properties  of  the  elements  and  their 
atomic  weights;  these  relations  are  applicable  to  all  the  elements 
and  have  the  nature  of  a  periodic  function."  This  important 
generalization  may  be  briefly  stated  thus:  The  properties  of  the 
elements  are  periodic  functions  of  their  atomic  weights. 

*  Chem.  News,  10,  94  (1864),  Ibid.,  12,  83  (1865). 
t  Lieb.  Ann.,  Suppl.,  8,  133  (1874). 


CLASSIFICATION  OF  THE  ELEMENTS  549 

The  original  table  of  Mendeleeff  has  been  amended  and  modi- 
fied as  new  data  has  accumulated  and  new  elements  have  been  dis- 
covered. The  accompanying  table,  though  containing  several 
new  elements,  and  an  entirely  new  group,  is  essentially  the  same  as 
that  of  Mendeleeff.  It  consists  of  nine  vertical  columns,  called 
groups,  and  twelve  horizontal  rows,  termed  series  or  periods.  The 
second  and  third  periods  contain  eight  elements  each,  and  are 
known  as  short  periods,  while  in  the  fourth  series,  starting  with 
argon,  it  is  necessary  to  pass  over  eighteen  elements  before  another 
element,  krypton,  is  encountered  which  bears  a  close  resemblance 
to  argon :  such  a  series  of  nineteen  elements  is  called  a  long  period. 
The  entire  table  is  composed  of  two  short  and  five  long  periods, 
the  last  one  being  incomplete.  The  positions  of  the  elements  are 
largely  determined  by  their  chemical  similarity  to  those  in  the 
same  group,  the  hyphens  indicating  the  positions  of  undiscovered 
elements.  The  elements  in  Group  VIII,  presented  difficulties 
when  Mendeleeff  attempted  to  place  them  according  to  their 
atomic  weights,  and  so  he  was  obliged  to  group  them  by  themselves. 
An  examination  of  the  table  shows,  that  the  valence  of  the  elements 
toward  oxygen  progresses  regularly  from  Group  O,  containing 
elements  which  exhibit  no  combining  power,  up  to  Group  VIII, 
where  it  attains  a  maximum  value  of  eight  in  the  case  of  osmium. 
The  valence  toward  hydrogen  on  the  other  hand  increases  regu- 
larly from  Group  VII  to  Group  IV,  in  which  the  elements  are 
quadrivalent. 

The  formulas  of  the  typical  oxides  and  hydrides  of  the  elements 
in  the  several  groups  are  indicated  at  the  top  of  each  vertical 
column  in  the  table,  where  R  denotes  any  element  in  the  group. 
The  valence  of  elements  in  the  long  periods  are  apt  to  be  variable. 
The  elements  in  the  second  series  are  frequently  called  bridge 
elements,  since  they  bear  a  closer  relation  to  the  elements  in  the 
next  adjacent  group  than  they  do  to  any  other  members  of  the 
same  group  in  succeeding  series.  The  members  of  the  third  series 
are  called  typical  elements,  because  they  exhibit  the  general  prop- 
erties and  characteristics  of  the  group.  Each  group  is  divided 
into  subgroups,  the  elements  on  the  right  and  left  sides  of  a  col- 
umn, forming  families,  the  members  of  which  are  more  closely 
related  than  are  all  of  the  elements  included  within  the  group. 
In  other  words,  we  detect  a  kind  of  periodicity  within  each 
group. 


550 


THEORETICAL  CHEMISTRY 


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CLASSIFICATION  OF  THE  ELEMENTS 


551 


In  any  given  series,  the  element  with  the  lowest  atomic  weight 
possesses  the  strongest  basic  character.  Thus,  we  find  the 
strongly  basic,  alkali  metals  on  the  left  side  of  the  table,  while 
on  the  right  side  are  the  acidic  elements  such  as  the  halogens  and 
sulphur.  In  fact,  the  strictly  non-metallic  elements  are  confined 
to  the  upper  right-hand  corner  of  the  table. 

Similarly,  as  we  pass  from  the  top  to  the  bottom  of  the  table, 
we  observe  a  progressive  change  in  the  base-forming  tendency  of 
the  elements,  that  is,  as  the  atomic  weight  increases,  the  metallic 
character  of  the  elements  in  each  group  becomes  more  pronounced. 

Periodicity  of  Physical  Properties.  Lothar  Meyer,  as  has  been 
pointed  out,  discovered  the  periodic  relations  of  the  elements  at 


40      «    50  60 

Atomic  Number 

Fig.  132 

about  the  same  time  as  Mendele"eff.  His  table  differed  but 
slightly  from  that  already  given.  The  most  important  part  of 
Meyer's  *  work,  however,  was  in  pointing  out  that  various  phys- 
ical properties  of  the  elements  are  periodic  functions  of  their 
atomic  weights.  We  know  to-day,  that  such  properties  as  specific 
gravity,  atomic  volume,  melting-point,  hardness,  ductility,  com- 
pressibility, thermal  conductivity,  coefficient  of  expansion,  spe- 
cific refraction,  and  electrical  conductivity  are  all  periodic.  When 
the  numerical  values  of  these  properties  are  plotted  as  ordinates 
*  Die  Modernen  Theorien  der  Chemie. 


552  THEORETICAL  CHEMISTRY 

against  their  atomic  weights,  or  atomic  numbers  (see  p.  587)  as 
abscissae,  we  obtain  wave-like  curves  similar  to  those  shown  in 
Fig.  132.  The  specific  heats  of  the  elements  are  an  exception  to 
the  general  rule.  According  to  the  law  of  Dulong  and  Petit,  the 
product  of  specific  heat  and  atomic  weight  is  a  constant,  and  con- 
sequently the  graphic  representation  of  this  relation  must  be  an 
equilateral  hyperbola. 

Applications  of  the  Periodic  Law.  Mendeleeff  pointed  out  the 
four  following  ways  in  which  the  periodic  law  could  be  employed: 
—  (1)  The  classification  of  the  elements;  (2)  The  estimation  of  the 
atomic  weights  of  elements;  (3)  The  prediction  of  the  properties 
of  undiscovered  elements;  and  (4)  The  correction  of  atomic 
weights. 

1.  Classification  of  Elements     The  use  of  the  periodic  law  in 
this  direction  has  already  been  indicated.     It  is  without  doubt 
the  best  system  of  c  assification  known,  and  is  to  be  ranked  among 
the  great  generalizations  of  the  science  of  chemistry. 

2.  Estimation   of  Atomic   Weights.     Because   of   experimental 
difficulties,  it  is  not  always  possible  to  -fix  the  atomic  weight  of  an 
element  by  determinations  of  the  vapor  densities  of  some  of  its 
compounds,  or  by  a  determination  of  its  specific  heat.     In  such 
cases,  the  periodic  law  has  proved  of  great  value.     An  historic 
example  is  that  of  indium,  the  equivalent  weight  of  which  was 
found  by  Winkler  to  be  37.8.     The  atomic  weight  of  the  element 
was  thought  to  be  twice  the  equivalent  weight  or  75.6.     If  this 
were  the  correct  value,  it  would  find  a  place  in  the  periodic  table 
between  arsenic  and  selenium.     Clearly,  there  is  no  vacancy  in 
the  table  at  this  point,  and  furthermore,  its  properties  are  not 
allied  to  those  of  arsenic  or  selenium.     Mendeleeff  proposed  to 
assign  to  it  an  atomic  weight  three  times  its  equivalent  weight,  or 
113.4,  when  it  would  fall  between  cadmium  and  tin  in  the  table. 
This  would  bring  it  in  the  same  group  with  aluminium,  the  typical 
element  of  the  group,  to  which  it  bears  a  close  resemblance.     This 
suggestion  of  MendeleefFs  was  confirmed  by  a  subsequent  deter- 
mination of  the  specific  heat  of  indium. 

3.  Prediction  of  Properties  of   Undiscovered  Elements.     At  the 
time  when  Mendeleeff  published  his  first  table,  there  were  many 
more  vacant  spaces  than  exist  in  the  present  periodic  table.     He 
ventured  to  predict  the  properties  of  many  of  these  unknown 
elements  by  means  of  the  average  properties  of  the  two  neighbor- 


CLASSIFICATION  OF  THE  ELEMENTS 


553 


ing  elements  in  the  same  series,  and  the  two  neighboring  elements 
in  the  same  sub-group.  These  four  elements  he  termed  atomic 
analogues.  The  undiscovered  elements  Mendeleeff  designated  by 
prefixing  the  Sanskrit  numerals,  eka  (one),  dwi  (two),  tri  (three), 
and  so  on,  to  the  names  of  the  next  lower  elements  of  the  sub- 
group. When  the  first  periodic  table  was  published,  there  were 
two  vacancies  in  Group  III,  the  missing  elements  being  called  by 
Mendeleeff  eka-aluminium  and  eka-boron,  while  in  Group  IV 
there  was  a  vacancy  below  titanium,  the  missing  element  being 
called  eka-silicon.  The  subsequent  discovery  of  gallium,  scan- 
dium and  germanium,  with  properties  nearly  identical  with  those 
predicted  for  the  above  hypothetical  elements,  served  to 
strengthen  the  faith  of  chemists  in  the  periodic  law.  The  follow- 
ing table  illustrates  the  accuracy  of  Mendeleeff 's  predictions. 
In  it  is  given  a  comparison  of  a  few  of  the  properties  of  the 


COMPARISON  OF  PROPERTIES  EKA-SILICON 
AND  GERMANIUM 


Eka-silicon,  Es. 


Germanium,  Ge. 


Atomic  weight,  72 

Specific  gravity,  5.5. 

Atomic  volume,  13. 

Metal  dirty  gray,  and  on  ignition 

yields  a  white  oxide,  EsO2. 
Element    decomposes    steam    with 

difficulty. 
Acids  have  slight  action,  alkalies  no 

pronounced  action. 


Action  of  Na  on  EsO2  or  on  EsK2F6 
gives  metal. 

The  oxide  EsO2  refractory. 

Specific  gravity  of  EsO2,  4.7. 

Basic  properties  of  EsO2  less  marked 
than  TiO2  and  SnO2,  but  greater 
than  SiO2. 

Fornis  hydroxide  soluble  in  acids, 
and  the  solutions  readily  decom- 
pose forming  a  metahydrate. 

EsCl4  a  liquid  with  a  b.p.  below  100° 
and  a  sp.  gr.  of  1.9  at  0°. 

EsF4  not  gaseous. 

Es  forms  a  compound  Es(C2H6)4  boil- 
ing at  160°,  and  with  a  sp.  gr.  0.96. 


Atomic  weight,  72.3. 
Specific  gravity,  5.47. 
Atomic  volume,  13.2. 
Metal  grayish-white,  and  on  igni- 
tion yields  a  white  oxide,  GeO2. 
Element  does  not  decompose  water. 

Metal  not  attacked  by  HC1,  but 
acted  upon  by  aqua  regia. 

Solutions  of  KOH  have  no  action. 
Oxidized  by  fused  KOH. 

Ge  obtained  by  reduction  of  GeOg 
with  C,  or  of  GeK2F6  with  Na. 

The  oxide  GeO2  refractory. 

Specific  gravity  of  GeO2,  4.703. 

Basic  properties  of  GeO2  feeble. 


Acids  do  not  ppt.  the  hydroxide 
from  dil.  alkaline  solutions,  but 
from  cone,  solutions,  acids  ppt. 
GeO  or  a  metahydrate. 

GeCl4  a  liquid  with  a  b.p.  of  86°, 
and  a  sp.  gr.  at  18°  of  1.887. 

GeF4.3  H2O  a  white  solid. 
Ge  forms  a  compound  Ge(C2Hs)< 
boiling  at  160°  and  with  a  sp.  gr. 
slightly  less  than  water. 


554  THEORETICAL  CHEMISTRY 

hypothetical  element,  eka-silicon,  as  predicted  by  Mendeleeff  in 
1871,  with  the  experimentally  determined  properties  of  germa- 
nium, discovered  by  Winkler  fifteen  years  later. 

4.  Correction  of  Atomic  Weights.  When  an  element  falls  in  a 
position  in  the  periodic  table  where  it  clearly  does  not  belong, 
suspicion  as  to  the  correctness  of  its  atomic  weight  is  immediately 
aroused.  Frequently  a  redetermination  of  the  atomic  weight  has 
revealed  an  error  which,  when  corrected,  has  resulted  in  assigning 
the  element  to  a  place  among  its  analogues.  Formerly,  the 
accepted  atomic  weights  of  osmium,  iridium,  platinum  and  gold 
were  in  the  order, 

Os  >  Ir  >  Pt  >  Au. 

But  from  analogies  existing  between  osmium,  ruthenium  and  iron 
and  the  disposition  of  the  preceding  members  of  Group  VIII, 
Mendeleeff  predicted  that  the  atomic  weights  were  in  error,  and 
that  the  order  of  the  elements  should  be, 

Os  <  Ir  <  Pt  <  Au. 

Subsequent  atomic  weight  determinations  by  Seubert,  substan- 
tiated MendeleefFs  prediction. 

Defects  in  the  Periodic  Law.  While  the  arrangement  of  the 
elements  in  the  periodic  table  is  on  the  whole  very  satisfactory, 
there  are  several  serious  defects  in  the  system  which  should  be 
pointed  out.  At  the  very  outset  there  is  difficulty  in  finding  a 
place  for  hydrogen  in  the  system.  The  element  is  univalent,  and 
falls  either  in  Group  I,  with  the  alkali  metals,  or  in  Group  VII  with 
the  halogens.  While  hydrogen  is  electro-positive,  it  cannot  be 
considered  to  possess  metallic  properties.  It  forms  hydrides  with 
some  of  the  metallic  elements,  and  can  be  displaced  by  the  halo- 
gens from  organic  compounds.  These  facts  make  it  extremely 
difficult  to  decide  whether  hydrogen  should  be  placed  in  Group  I 
or  Group  VII.  The  idea  has  been  advanced,  that  hydrogen  is 
the  only  known  member  of  the  first  series  of  the  periodic  table. 
These  hypothetical  elements  have  been  called  proto-elemente, 
the  successive  members  of  the  series  being,  proto-glucinum,  proto- 
boron  and  so  on,  to  the  last  element  in  the  series,  proto-fluorine. 

To  find  a  suitable  location  for  the  rare-earth  elements  in  the  peri- 
odic system  is  another  difficulty  which  has  .not  been  satisfactorily 
met.  Brauner  considers  that  these  elements  should  all  be  grouped 
together  with  cerium  (at.  wt.  =  140.25),  but,  owing  to  our  limited 


CLASSIFICATION  OF  THE  ELEMENTS  555 

knowledge  of  the  properties  of  these  elements,  it  seems  better  to 
defer  attempting  to  place  them  for  the  present. 

In  the  group  of  non-valent  elements,  the  atomic  weight  of  argon 
is  distinctly  higher  than  that  of  potassium  in  the  next  group. 
There  can  be  little  doubt  that  the  values  of  their  atomic  weights 
are  correct,  and  it  is  evidently  impossible  to  interchange  the  posi- 
tions of  these  two  elements  in  the  periodic  table,  since  argon  is  as 
much  the  analogue  of  the  rare  gases,  as  potassium  is  of  the  alkali 
metals.  A  similar  discrepancy  occurs  with  the  elements,  tellurium 
and  iodine.  The  atomic  weight  of  the  former  element  is  ap- 
preciably higher  than  that  of  the  latter  and,  notwithstanding  the 
attempts  of  numerous  investigators  ,to  prove  tellurium  to  be  a 
complex  of  two  or  more  elements,  nothing  but  failure  has  attended 
their  efforts.  Still  another  anomaly  is  enjountered  in  Group  VII, 
where  manganese  is  classed  with  the  halogen  family,  to  which 
it  bears  much  less  resemblance  than  it  does  to  chromium  and  iron, 
its  two  immediate  neighbors. 

As  has  already  been  mentioned,  Group  VIII  is  made  up  of  non- 
conformable  elements.  If  the  properties  of  the  elements  are 
dependent  upon  their  atomic  weights,  it  should  be  impossible  for 
several  elements  having  almost  identical  atomic  weights  and 
different  properties  to  exist,  and  yet  such  is  the  case  with  the 
elements  of  Group  VIII.  The  elements  copper,  silver  and  gold, 
while  not  closely  resembling  the  other  members  of  Group  VIII, 
are  much  more  closely  allied  to  them  than  to  the  alkali  metals 
with  which  they  are  also  classed. 

In  the  light  of  recent  discoveries,  the  defects  in  the  periodic 
system  are  seen  to  be  more  apparent  than  real,  for  it  has  been 
shown  that  there  is  a  property  known  as  the  "  atomic  number  " 
which  is  even  more  fundamental  than  atomic  weight,  and  when  the 
elements  are  arranged  according  to  this  property  the  irregulari- 
ties in  the  system  disappear.  Before  discussing  the  periodic 
system  from  the  standpoint  of  atomic  numbers,  however,  it  will 
be  necessary  to  give  a  brief  outline  of  the  series  of  brilliant  experi- 
mental researches  which  have  led  to  the  modern  viewpoint  as  to 
atomic  structure. 

REFERENCES 

The  Periodic  Law.     Venable. 
The  Elements.     Tilden. 


CHAPTER  XXI 
THE  ELECTRICAL  THEORY   OF  MATTER 

Conduction  of  Electricity  through  Gases.  Within  recent 
years  the  discovery  of  new  facts  relative  to  the  conduction  of 
electricity  through  gases  has  led  to  the  development  of  the  so- 
called  electron  or  corpuscular  theory  of  matter.  Under  ordinary 
conditions,  gases  are  practically  non-conductors  of  electricity,  but 
when  a  sufficiently  great  difference  of  potential  is  established 
between  two  points  within  a  gas,  it  is  no  longer  able  to  withstand 
the  stress,  and  an  electric  discharge  takes  place  between  the  points. 
The  potential  necessary  to  produce  such  a  discharge  is  quite  high, 
several  thousand  volts  being  required  to  produce  a  spark  of  one 
centimeter  length  in  air  at  ordinary  pressures.  The  pressure  of 

the  gas  has  a  marked  effect  upon 
the  character  of  the  discharge 
and  the  potential  required  to 
produce  it.  If  we  make  use  of  a 
glass  vessel  similar  to  that  shown 
in  Fig.  133,  the  effect  of  pressure 
t  TO  pump  on  the  nature  Of  the  discharge 

may  be  studied.  This  ap- 
paratus consists  of  a  straight  glass  tube  about  4  cm.  in  diameter 
and  40  cm.  long,  into  the  ends  of  which  platinum  electrodes 
are  sealed.  To  the  side  of  the  vessel,  a  small  tube  is  sealed 
so  that  connection  may  be  established  with  an  air-pump  and 
manometer.  If  the  electrodes  are  connected  with  the  terminals 
of  an  induction  coil,  and  the  pressure  within  the  tube  be  gradually 
diminished,  the  following  changes  in  the  character  of  the  dis- 
charge will  be  observed.  At  first,  the  spark  becomes  more  uni- 
form and  then  broadens  out,  assuming  a  bluish  color.  When  a 
pressure  of  about  0.5  mm.  is  reached,  the  negative  electrode,  or 
cathode  will  appear  to  be  surrounded  by  a  thin  luminous  layer;  next 
to  this  will  be  found  a  dark  region,  known  as  the  Crookes'  dark  space; 
adjoining  this  will  be  found  a  luminous  portion,  called  the  negative 
glow,  and  beyond  this  will  be  seen  another  dark  region,  which  is 

556 


THE  ELECTRICAL  THEORY  OF  MATTER      557 

frequently  referred  to  as  the  Faraday  dark  space.  Between  the 
Faraday  dark  space  and  the  positive  electrode,  or  anode  is  a  lumi- 
nous portion,  called  the  positive  column.  By  a  slight  variation  of 
the  current  and  pressure,  the  positive  column  can  be  caused  to 
break  up  into  alternate  light  and  dark  spaces  or  strice,  the  appear- 
ance of  which  is  dependent  upon  various  factors,  such  as  the  nature 
of  the  gas  and  the  size  of  the  tube.  If  the  pressure  in  the  tube 
be  diminished  to  about  0.01  mm.,  a  new  phenomenon  will  be  ob- 
served. The  positive  column  will  vanish,  and  the  walls  of  the 
tube  opposite  the  cathode  will  become  faintly  phosphorescent. 
The  color  of  the  phosphorescence  will  depend  upon  the  nature  of 
the  glass.  If  the  tube  is  made  of  soda  glass,  the  glow  will  be  green- 
ish yellow,  while  with  lead  glass  the  phosphorescence  will  be  blu- 
ish. The  phosphorescence  is  due  to  the  bombardment  of  the 
walls  of  the  tube  by  very  minute  particles  projected  normally 
from  the  cathode.  These  streams  of  particles  are  called  the 
cathode  rays. 

Some  Properties  of  Cathode  Rays.     The  following  are  among 
the  most  important  properties  of  the  cathode  rays: 

1.  The  cathode  rays  travel  in  straight  lines  normal  to  the  cathode, 
and  cast  shadows  of  opaque  objects  placed  in  their  path.      This 
property  may  be  demonstrated  by  means  of  a  vacuum  tube  in 
which  a  small  metallic  Maltese  cross  is  interposed  in  the  path  of 
the  rays.     When  the  current  passes,  a  distinct  shadow  of  the  cross 
is  cast  on  the  opposite  wall  of  the  tube.     The  cross  may  be  hinged 
at  the  bottom  so  that  it  can  be  dropped  out  of  the  path  of  the  rays, 
when  the  usual  phosphorescence  will  be  obtained. 

2.  The  cathode  rays  can  produce  mechanical  motion.     By  means 
of  an  apparatus  devised  by  Sir  William  Crookes,  this  property 
of  the  cathode  rays  may  be  demonstrated.     Within  a  vacuum 
tube  is  placed  a  small  paddle  wheel  which  rolls  horizontally  on  a 
pair  of  glass  rails.     When  the  current  is  applied  to  the  tube,  the 
wheel  will  revolve,  moving  away  from  the  cathode.     By  reversing 
the  current,  the  wheel  will  stop,  and  then  rotate  in  the  opposite 
direction,  owing  fo  the  reversal  of  the  direction  of  the  cathode 
stream. 

3.  The  cathode  rays  cause  a  rise  of  temperature  in  objects  upon 
which  they  fall.     If  a  small  piece  of  platinum  is  placed  at  the  center 
of  curvature  of  a  spherical  cathode  in  a  vacuum  tube,  and  a  strong 
discharge  be  sent  through  the  tube,  the  anode  will  begin  to  glow, 


558  THEORETICAL  CHEMISTRY 

and  if  the  action  of  the  current  be  continued  long  enough,  the 
platinum  plate  may  be  rendered  incandescent,  thus  showing  the 
marked  heating  effect  of  the  cathode  rays. 

4.  Many  substances  become  phosphorescent  on  exposure  to  the 
cathode  rays.     If  the  cathode  rays  be  directed  upon  different  sub- 
stances,   such   as   calc-spar,    barium    platino-cyanide,    willemite, 
scheelite  and  various  kinds  of  glass,   beautiful  phosphorescent 
effects  may  be  observed.     This  phosphorescent  property  is  use- 
ful in  observing  and  experimenting  with  the  cathode  rays. 

5.  The  cathode  rays  can  be  deflected  from  their  rectilinear  path  by 
a  magnetic  field.     If  a  phosphorescent  screen  is  placed  opposite 
to  the  cathode  in  a  vacuum  tube,  a  circular  phosphorescent  spot 
will  appear  when  the  tube  is  excited.     If  the  tube  be  placed  be- 
tween the  poles  of  an  electromagnet,  the  phosphorescent  spot  will 
move  at  right  angles  to  the  direction  of  the  magnetic  field.     On 
reversing  the  polarity  of  the  magnet,  the  spot  will  move  in  the 
opposite  direction.     Furthermore,  the  direction  of  the  deflection 
will  be  found  to  be  similar  to  that  produced  by  a  negative  charge 
of  electricity  moving  in  the  same  direction  as  the  cathode  ray. 

6.  The  cathode  rays  can  be  deflected  from  their  rectilinear  path 
by  an  electrostatic  field.     If  two  insulated  metal  plates,  are  sealed 
into  the  sides  of  a  vacuum  tube,  parallel  and  opposite  to  each  other, 
and  a  difference  of  potential  of  several  hundred  volts  be  applied 
to  the  plates,  the  luminous  spot  produced  by  the  rays  on  a  phos- 
phorescent screen  will  be  found  to  move,  the  direction  of  the 
motion  being  the  same  as  that  of  a  negatively  charged  body  under 
the  influence  of  an  electrostatic  field.     Reversal  of  the  field,  causes 
the  phosphorescent  spot  to  move  to  the  opposite  side  of  the  screen. 

7.  The  cathode  rays  carry  a  negative  charge.     Probably  the  most 
important  characteristic  of  the  cathode  rays  is  their  ability  to 
carry  a  negative  charge.     While  the  magnetic  and  electrostatic 
deviation  of  the  rays  made  this  fact  more  than  probable,  it  re- 
mained for  Perrin  to  demonstrate,  that  a  negative  electrification 
accompanies   the   cathode   stream      A   modification   of   Perrin's 
apparatus,  due  to  Sir  J.  J.  Thomson,  is  shown  in  Fig.  134.     It  con- 
sists of  a  spherical  bulb  to  which  is  sealed  a  smaller  bulb,  and  a  long 
side  tube.     The  small  bulb  contains  the  cathode,  C,  and  the  anode, 
A.     The  anode  consists  of  a  tight-fitting  brass  plug  pierced  by  a 
central  hole  of  small  diameter.     The  side  tube,  which  is  out  of 
the  direct  range  of  the  cathode  rays,  contains  two  coaxial  metallic 


THE  ELECTRICAL  THEORY  OF   MATTER 


559 


cylinders,  insulated  from  each  other,  each  being  perforated  with  a 
narrow  transverse  slit.  One  of  these,  D,  is  earth-connected,  while 
B  is  connected  with  an  electrometer,  by  means  of  the  rod,  F.  When 
the  tube  has  been  pumped  down  to  the  proper  pressure  for  the  pro- 
duction of  cathode  rays,  a  phosphorescent  spot  will  appear  at  E, 


Elect; 


Fig.  134 


To  Pump 


directly  opposite  the  cathode,  C.  Upon  testing  B  for  possible  elec- 
trification by  means  of  the  electrometer,  it  will  be  found  to  be  un- 
charged. If  the  cathode  stream  be  deflected  by  means  of  a  magnet, 
so  that  the  rays  fall  upon  B,  a  sudden  charging  of  the  electrometer 
will  be  observed,  proving  that  B  is  becoming  electrified.  Upon 
deflecting  the  rays  still  further,  so  that  they  are  no  longer  incident 
upon  B,  the  accumulation  of  the  charge  will  immediately  cease.  If 
the  electrometer  be  tested  for  polarity,  it  will  be  found  to  be  nega- 
tively charged,  thus  proving  the  charge  carried  by  the  cathode 
rays  to  bo  negative. 

8.  The  cathode  rays  can  penetrate  thin  sheets  of  metal.     In  1894, 
Lenard  constructed  a  vacuum  tube  fitted  with  an  aluminium 
window  opposite  the  cathode.     He  showed,  that  the  cathode  rays 
passed  through  the  aluminium,  and  are  absorbed  by  different  sub- 
stances outside  of  the  tube,  the  absorption  varying  directly  with 
the  density  of  the  substance 

9.  The  cathode  rays  when  dire  ted  into  moist  air  cause  the  forma- 
tion of  fog.     This  phenomenon  has  been  shown  by  C.  T.  R.  Wilson 
to  be  due  to  the  minute  particles  in  the  cathode  stream  acting  as 
nuclei  upon  which  the  water  vapor  can  condense. 


560  THEORETICAL  CHEMISTRY 

Velocity  of  the  Cathode  Particle.  Since  the  cathode  rays 
consist  of  minute,  negatively-charged  particles  which  can  be 
deflected  by  a  magnetic  and  an  electrostatic  field,  it  is  possible 
to  measure  their  speed,  and  to  compute  the  ratio  of  the  mass  of  a 
particle  to  its  charge.  The  special  form  of  tube  shown  in  Fig. 
135  was  devised  for  the  purpose  by  Sir  J.  J.  Thomson.  It  consists 
of  a  glass  tube  about  60  cm.  in  length,  furnished  with  a  flat  cir- 


Fig.  135 

cular  cathode,  C,  and  an  anode,  A,  in  the  form  of  a  cylindrical 
brass  plug  about  2.5  cm.  in  length,  pierced  by  a  central  hole  1  mm. 
in  diameter.  Another  brass  plug,  B,  is  placed  about  5  cm.  away 
from  A)  the  two  holes  being  in  exactly  the  same  straight  line,  so 
that  a  very  narrow  bundle  of  rays  may  pass  along  the  axis  of  the 
tube  and  fall  upon  the  phosphorescent  screen  at  the  opposite 
end  of  the  tube.  Upon  this  screen  is  a  millimeter  scale,  SS'. 
Two  parallel  plates,  D  and  E,  are  sealed  into  the  tube  for  the  pur- 
pose of  establishing  an  electrostatic  field.  When  the  tube  is 
connected  with  an  induction  coil,  or  other  source  of  high-potential, 
a  phosphorescent  spot  will  appear  at  F.  If  a  strong  magnetic 
field  be  applied,  the  lines  of  force  being  at  right  angles  to  the 
plane  of  the  diagram,  the  rays  will  be  deflected  vertically, 
and  the  spot  on  the  screen  will  move  from  F  to  G.  Let  H 
denote  the  strength  of  the  magnetic  field,  and  let  m,  e  and 
v  represent,  respectively,  the  mass,  charge  and  velocity  of  a 
cathode  particle.  A  magnetic  field,  H,  acting  at  right  angles  to 
the  line  of  flight  of  the  cathode  particle  will  exert  a  force,  Hev, 
which  will  tend  to  deflect  the  particle  from  a  rectilinear  path. 
This  force  must  be  equal  to  the  centrifugal  force  of  the  moving 
particle  acting  outwards  along  its  radius  of  curvature.  Therefore, 

mv* 

Hev  =  — > 
r 


THE  ELECTRICAL  THEORY  OF  MATTER  561 

Or  rr  mV 

Hr  =  ^'  (1) 

Since  H  and  r  can  both  be  measured,  the  ratio,  —  ,  can  be  deter- 
mined. Now  if  a  difference  of  potential  be  established  between 
D  and  E,  and  the  lines  of  force  in  the  electrostatic  field  have  the 
proper  direction,  it  will  be  possible  to  alter  the  strength  of  the  field 
so  as  just  to  counterbalance  the  effect  of  the  magnetic  field,  and 
bring  the  phosphorescent  spot  back  to  F  again.  Under  these 
conditions,  if  X  denotes  the  strength  of  the  electrostatic  field,  we 
have 

Xe  =  Hev, 

or  X 

v  =  jj  •  (2) 

Since  X  and  H  can  both  be  measured,  v  can  be  calculated,  and,  by 
introducing  the  value  so  obtained  into  equation  (1),  the  ratio 
e/m  can  be  evaluated.  By  this  method,  the  average  value  of  v 
has  been  found  to  be  2.8  X  109  cm.  per'second,  while  5.30  X  10~17 
electrostatic  units  is  the  mean  value  of  the  most  trustworthy 
determinations  of  the  ratio,  e/m. 

Charge  Carried  by  the  Cathode  Particle.  Until  the  value 
of  the  charge  carried  by  the  cathode  particle  has  been  determined, 
it  is  clearly  impossible  to  compute  its  mass.  Only  a  brief  out- 
line of  the  method  employed  to  determine  this  quantity  can  be  given 
here.  Upon  suddenly  expanding  a  volume  of  saturated  water  vapor, 
its  temperature  is  lowered  with  the  simultaneous  formation  of  a 
cloud,  each  particle  of  dust  present  serving  as  a  nucleus  for  a  fog 
particle.  If  sufficient  time  be  allowed  for  the  mist  to  settle  and  the 
vapor  to  become  saturated  again,  a  repetition  of  the  preceding  proc- 
ess will  result  in  the  formation  of  less  mist,  owing  to  the  presence  of 
fewer  dust  particles.  By  repeating  the  operation  enough  times,  the 
space  may  be  rendered  dust  free.  As  has  already  been  pointed 
out,  cathode  particles  serve  as  nuclei  for  the  condensation  of 
water  vapor,  their  function  being  similar  to  that  of  dust  particles. 
It  has  been  shown  by  Sir  George  G.  Stokes,  that  if  a  drop  of  water 
of  radius  r,  be  allowed  to  fall  through  a  gas  of  viscosity  ij,  then  the 
velocity  with  which  the  drop  falls  will  be  given  by  the  equation, 


562  THEORETICAL  CHEMISTRY 

where  g  is  the  acceleration  due  to, gravity.  The  viscosity  of  air 
at  any  temperature  being  known,  a  cloud  can  be  produced  by  ex- 
pansion of  water  vapor  in  the  presence  of  cathode  particles  in  an 
appropriate  chamber.  The  speed,  v,  with  which  the  cloud  falls 
can  be  measured,  and  hence  r  can  be  calculated  by  means  of  equa- 
tion (3) .  If  m  is  the  total  mass  of  the  cloud,  and  n  is  the  number  of 
drops  per  cubic  centimeter,  then 

m  =  4/3  mrr3  (density  of  water  =  1) . 

A  simple  application  of  thermodynamics  enables  one  to  calculate 
the  value  of  m.  Knowing  the  values  of  m  and  r,  the  number  of 
drops  in  the  cloud,  n,  which  is  the  same  as  the  number  of  cathode 
particles,  can  be  calculated.  It  is  a  simple  matter  to  measure  the 
total  charge  in  the  expansion  chamber,  and  if  this  be  divided  by  the 
total  number  of  charged  particles,  the  charge  carried  by  a  single  par- 
ticle will  be  obtained.  The  latest  determinations  show  this  to  be 
4.774  X  10~10  electrostatic  unit.  This  being  practically  identical 
with  the  calculated  value  of  the  charge  on  the  hydrogen  ion  in  elec- 
trolysis, it  follows,  that  the  mass  of  the  cathode  particle  is  1845 
times  less  than  the  mass  of  the  atom  of  hydrogen,  or  0.00054  on 
the  scale  of  atomic  weights  in  which  0  =  16.  The  cathode  par- 
ticle has  the  smallest  mass  yet  known  and  has  been  called  the 
corpuscle  or  lectron. 

An  ingenious  modification  of  the  foregoing  method,  based  upon 
the  fact  that  when  X-rays  are  passed  through  air  or  any  other  gas 
some  of  the  molecules  are  broken  up  into  positively  or  negatively 
charged  carriers  of  electricity  called  ions,  has  enabled  Millikan  * 
to  determine  the  value  of  e  with  extreme  accuracy.  A  sketch 
of  the  apparatus  used  by  him  is  shown  in  the  accompanying  illus- 
tration, Fig.  136.  A  finely  divided  spray  of  oil,  or  some  other 
liquid  is  introduced  into  the  chamber,  D,  by  means  of  the  atom- 
izer, A ,  until  one  of  the  tiny  drops  eventually  finds  its  way  through 
the  aperture,  p,  into  the  space  between  the  two  condenser  plates, 
M  and  N.  The  hole,  p,  is  then  closed  and  the  air  between  the 
plates  is  ionized  by  means  of  the  X-ray  tube,  X.  The  arc  lamp, 
a,  serves  to  illuminate  the  drop,  and  its  motion  is  observed  through 
a  microscope,  not  shown  in  the  figure.  The  eye-piece  of  the  micro- 
scope is  fitted  with  an  engraved  glass  scale,  which  enables  the 
observer  to  measure  accurately  the  motion  of  the  drop.  When 

*  Phys.  Rev.  39,  349  (1911);  2,  143  (1913). 


THE  ELECTRICAL  THEORY  OF  MATTER 


563 


the  drop  collides  with  one  of  the  ions  of  the  air  in  the  condenser 
chamber,  it  acquires  an  electric  charge,  the  sign  of  which  will 
determine  the  subsequent  direction  of  its  motion  with  reference 
to  M  and  N.  By  means  of  the  switch,  S,  the  signs  of  the 
charges  on  the  two  plates  can  be  reversed,  thus  causing  a  corre- 


Fig.  136 

spending  reversal  of  the  direction  of  motion  of  the  drop.  By 
careful  adjustment  of  the  difference  of  potential  between  the 
plates,  it  is  possible  to  keep  the  drop  in  the  field  of  view  during 
an  entire  experiment.  Any  change  in  the  speed  of  the  drop  is  an 
indication  of  a  change  in  the  charge  which  it  carries.  Therefore, 
since  the  speed  with  which  the  drop  moVes  is  directly  proportional 
to  the  magnitude  of  its  charge,  it  follows,  that  by  measuring  the 
change  of  speed  and  comparing  it  with  the  speed  of  the  drop  when 
falling  under  the  influence  of  gravitational  attraction  alone,  the 
value  of  the  charge  carried  by  the  drop  can  be  calculated. 

In  this  manner  Millikan  found,  that  the  total  charge  on  the 
drop  was  invariably  an  exact  integral  multiple  of  the  elemental 
electronic  charge,  e.  It  was  observed,  that  seven  or  eight  elec- 
tronic charges  were  quite  commonly  acquired  by  a  single  drop, 
but  in  every  case  the  total  charge  was  invariably  an  exact  integral 
multiple  of  the  elemental  charge,  e.  The  value  of  e  given  by 
Millikan  as  the  mean  of  a  large  number  of  determinations  is  4.774 
X  10~10  electrostatic  units:  this  result  is  believed  to  be  in  error 
by  less  than  0.1  percent.  In  the  following  table  are  given  some 
of  the  more  recently  determined  values  of  e,  as  found  by  several 
observers  using  different  experimental  methods. 


564  THEORETICAL  CHEMISTRY 

VALUES  OF  THE  ELEMENTAL  ELECTRIC  CHARGE,  e 


Observer 

Method 

e  in  e.s.u. 

Millikan  

Oil  or  mercury 

4  77  X  10-10 

Begeman  ...    . 

Water  drops 

4  67  X  10-10 

Roux 

Sulphur  drops 

4  17  X  10~10 

Ehrehaft 

Metallic  dust 

4  65  X  10-10 

Rutherford.  . 

Counting  a-particles 

4  65  X  10-10 

Planck  

Radiation  data 

4  69  X  10-10 

Avogadro's  Constant.  The  ratio  of  mass  to  charge  for  the 
hydrogen  ion  in  electrolysis  is  known  to  be  1.04  X  10"4  and,  as 
we  have  seen,  the  value  of  the  charge  is  identical  with  that  carried 
by  the  electron  viz.,  4.774  X  10~10  electrostatic  units,  or  1.57  X 
10~20  electromagnetic  units.  It  follows,  therefore,  that  the  mass, 
m,  of  the  hydrogen  atom  must  be,  1.04  X  10-4  X  1.57  X  lO"20  = 
1.63  X  10~24  grams.  Having  established  the  weight  of  the  hy- 
drogen atom  in  grams,  the  weight  in  grams  of  the  atom  of  any 
other  element  can  be  calculated  from  the  ratio  of  its  atomic  weight 
to  the  atomic  weight  of  hydrogen.  The  mass  of  one  cubic  centi- 
meter of  hydrogen,  measured  under  standard  conditions  of  tem- 
perature and  pressure,  is  8.81  X  10~5  grams;  hence  the  number 
of  atoms  present  in  one  cubic  centimeter  of  hydrogen,  under 
standard  conditions,  is  8.81  X  10~5  -5-  1.63  X  10~24  =  5.41  X  1019 
grams.  Since  the  molecule  of  hydrogen  is  diatomic,  it  follows 
that  the  number  of  molecules  contained  in  one  cubic  centimeter 
must  be  5.41  X  1019  -^2  =  2T.705  X  1019.  But  one  gram-mole- 
cule of  any  gas,  when  measured  under  standard  conditions,  occu- 
pies 22400  cc.;  therefore,  the  number  of  molecules  present  in 
one  gram-molecule  is,  22400  X  2.705  X  1019  =  6.06  X  10™.  This 
fundamental  quantity  is  known  as  Avogadro's  Constant.  A  table 
of  the  values  of  this  constant,  as  determined  by  different  ob- 
servers using  different  methods,  is  given  on  p.  24. 

Positive  Rays.  As  we  have  seen,  the  cathode  rays  consist  of 
negatively  charged  particles  which  are  repelled  from  the  cathode 
of  a  vacuum  tube.  These  particles,  or  electrons,  under  the  in- 
fluence of  a  high  potential,  move  with  an  enormous  velocity  and, 
by  collision  with  the  atoms  of  the  residual  gas  within  the  tube, 
they  remove  one  or  more  electrons  from  these  atoms,  leaving  them 
with  a  positive  charge.  These  positively-charged  atoms  are 


THE  ELECTRICAL  THEORY  OF  MATTER      565 

immediately  repelled  by  the  anode,  and  move  with  high  velocity 
toward  the  cathode.  If  a  tube  with  a  perforated  cathode  be  used, 
these  positively  charged  atoms  will  pass  through  the  perforations 
into  the  space  behind  the  cathode,  where  their  properties  may  be 
studied.  These  rays  have  been  called  positive  rays  by  Sir  J.  J. 
Thomson,  who  has  developed  an  ingenious  method  for  the  deter- 
mination of  the  individual  masses  of  the  positively  charged  par- 
ticles constituting  the  rays.  While  this  method  of  analysis  of 
positive  rays  has  proved  of  great  value  for  a  general  survey  of 
masses  and  velocities,  it  leaves  much  to  be  desired  as  a  method 
of  precision. 

The  defects  in  Thomson's  method  have  been  overcome,  in  large 
measure,  by  Aston  *  who  has  recently  perfected  ah  apparatus 
for  positive-ray  analysis  which  he  calls  a  "  mass  spectrograph." 
This  instrument  is  illustrated  diagrammatically  in  Fig.  137.  The 


Fig.  137 


positive  rays  after  reaching  the  cathode,  pass  through  two  nar- 
row parallel  slits  of  special  construction,  SiSz,  which  permit  a 
thin  ribbon  of  rays  to  pass  between  the  charged  plates,  PI  and 
P2,  where  they  are  spread  out  into  an  electric  spectrum  under  the 
influence  of  the  electric  field.  After  emerging  from  the  electric 
field,  the  rays  may  be  considered  as  radiating  from  a  virtual  source 
Z,  midway  through  the  field  on  the  line,  $i$2.  A  portion  of  the 
emergent  rays,  deflected  through  an  angle,  6,  from  their  initial 
direction  is  sorted  out  by  means  of  a  diaphragm,  D,  and  allowed  to 
pass  between  the  parallel  poles  of  a  powerful  electro-magnet,  0. 
The  strength  of  the  magnetic  field  is  such,  that  the  rays  are  de- 
flected backward  through  an  angle,  </>,  which  is  more  than  twice  as 
great  as  6.  By  this  means,  all  of  the  rays  having  a  given  ratio 
of  charge  to  mass,  are  brought  to  a  focus,  F,  where  an  image  is 

*  Phil.  Mag.,  38,  709  (1919). 


566  THEORETICAL  CHEMISTRY 

produced  on  a  photographic  plate,  GF.  In  this  way,  each  class 
of  charged  particles  will  produce  its  own  image  at  its  own  partic- 
ular focus.  The  resulting  series  of  images  is  known  as  a  mass 
spectrum.  By  measuring  the  displacement  of  the  image  produced 
by  a  particle  of  unknown  mass,  and  comparing  this  with  the  corre- 
sponding displacement  of  the  image  produced  by  a  particle  whose 
mass  is  known,  it  is  possible  to  determine  the  mass  of  the  unknown 
particle.  Since  in  a  mass  spectrum,  all  measurements  are  rela- 
tive, any  known  element  may  be  selected  as  a  standard.  Ob- 
viously, oxygen  is  taken  as  the  standard,  its  molecule,  its  singly- 
charged  atom,  and  its  doubly-charged  atom  giving  reference  lines 
at  32,  16  and  8,  respectively.  In  a  mass  spectrum,  the  displace- 
ment with  increasing  mass  is  very  nearly  linear,  hence,  if  the  dis- 
tances from  a  chosen  point  of  reference  are  d\  and  d2,  and  if  mi 
and  mz  are  the  corresponding  masses,  it  follows,  that  for  given 
values  of  di  and  ch,  the  value  of  the  ratio,  nii/m*,  will  be  the  same 
in  every  experiment.  It  is  estimated  that  atomic  weights  can  be 
determined  in  this  manner  with  an  accuracy  of  0.1  per  cent. 
Small  amounts  of  carbon  monoxide  and  carbon  dioxide  are  fre- 
quently mixed  with  the  gas  under  investigation  in  order  to  pro- 
mote the  smooth  running  of  the  discharge  tube,  while  traces  of 
hydrocarbons,  derived  from  stop-cock  lubricants,  are  almost  in- 
variably present. 


Lntor-co    °     **••     ^r  m  «r>  r-  oo  <r>  o       cvi                                       o 

l£>i°!-r:5      CVI         CJ        OJ  <NJ  CJ  OJ  c\4  cvj  10          m                                                                 •*- 

I  riBii  in  u    '         '           '          i      i      i     i      i      I     i           I 

1    HI  II    11    II    III  111  111  III  II!  Ill                              1 

1!              INe 

UT>         CO 

ro       ro 

i  TTTI             1  1  in  in  i 

11   ill    ki 

CD                          oo            cxj        ir><or~-oo                        •«a- 

OJ                                     OJ                 MO           rONOhOhO                                 ^ 

i    i     ii  mi  nil  1  I    ii  M  iii     1     ii  mi 

lei 

i  in   Till    I    iii          Inii  I 

lei 

Fig.  138 

A  reproduction  of  the  mass  spectra  of  neon  and  chlorine 
is  given  in  Fig.  138.  The  relative  masses,  taking  oxygen  as  a 
standard,  are  indicated  by  the  accompanying  numbers.  The 
prominent  lines  in  the  neon  spectrum  correspond  to  0  =  16,  C2  = 
24,  C2H2  =  26  and  C2H4  or  CO  =  28,  while  the  two  lines  pro- 
duced by  neon  itself  are  found  to  correspond  to  atomic  weights  20.0 
and  22.0,  rather  than  to  20.2,  the  commonly  accepted  atomic 
weight  of  neon.  From  this  result,  Aston  concludes  that  neon 


THE  ELECTRICAL  THEORY  OF   MATTER 


567 


is  made  up  of  two  species  of  atoms  differing  in  atomic  weight  by 
two  units.  A  mixture  of  nine  parts  by  weight  of  the  lighter  atoms 
with  one  part  by  weight  of  the  heavier  atoms,  would  give  a  gas 
whose  atomic  weight  would  correspond  to  that  assigned  to  neon 
in  the  table  of  atomic  weights. 

In  like  manner,  the  mass  spectrum  of  chlorine  shows  intense 
lines  at  16,  28  and  44 :  these  are  undoubtedly  due  to  the  presence 
of  oxygen,  carbon  monoxide  and  carbon  dioxide  respectively,  while 
the  strong  lines  at  35.0,  36.0,  37.0,  and  38.0  are  believed  to  indi- 
cate the  existence  of  two  different  species  of  chlorine  atoms,  one  of 
which  has  an  atomic  weight  of  35.0  and  the  other  an  atomic  weight 
of  37.0.  The  other  two  lines  at  36.0  and  38.0  are  attributed  to 
the  presence  of  the  corresponding  hydrogen  compounds,  HC1' 
and  HC1",  where  Cl'  =  35.0  and  Cl"  =  36.0. 

As  will  be  pointed  out  in  the  following  chapter,  the  somewhat 
revolutionary  idea  that  substances  can  exist  with  practically 
identical  chemical  and  spectroscopic  properties  and  yet  differ  in 
atomic  weight,  was  advanced  by  Soddy  and  others,  several  years 
before  Aston  furnished  direct  experimental  evidence  of  the  fact. 
In  attempting  to  find  a  place  for  the  radio-elements  in  the  periodic 
table,  Soddy  found  it  necessary  to  assign  certain  elements  to  the 
same  position  in  the  table,  and  hence  proposed  that  such  elements' 
should  be  called  isotopes.  The  following  table  contains  a  list 
of  the  elements  and  their  isotopes  as  given  by  Aston.* 

ATOMIC  WEIGHTS  OF  ELEMENTS  AND  ISOTOPES 


Element 

At.  Wt. 

No.  of 
Isotopes 

At.  Wt.  of  Isotopes 

H 

1  008 

1 

1  .008 

He.. 

4.00 

1 

4 

Li. 

6.94 

2 

B.  ....... 

10.9 

2 

11,  10 

Mg.. 

24.36 

3 

24,  25,  26 

Si 

28  3 

2 

28,  29 

Cl 

35  46 

2 

35,  37 

A  ...     . 

39.9 

2 

36,  40 

K   . 

39.10 

2 

39,  41 

Ca  

40.07 

2 

40,  44 

Ni  

58.68 

2 

58,  60 

Zn  
Br  
Kr...      . 

65.37 
79.92 
82.92 

4 
2 
6 

64,  66,  68,  70 
79,81 
78,  80,  82,  83,  84,  86 

Rb  

85.45 

2 

85,  87 

Xe  

130.2 

7 

128,  129,  130,  131,  132,  134,  136 

Hg.. 

200.6 

6 

197,  198,  199,  200,  202,  204 

*  Aston,  Isotopes,  p.  142. 


568  THEORETICAL  CHEMISTRY 

It  is  apparent  from  the  foregoing  table,  that  the  isotopic  nature  of 
many  of  the  more  common  elements  has  been  fully  demonstrated. 
In  addition  to  the  positive-ray  method,  already  outlined,  several 
other  methods  have  been  developed  for  the  separation  of  isotopes. 
Of  these,  we  may  mention  the  methods  based  upon  diffusion,* 
fractional  distillation,!  and  free  evaporation,  {  each  of  which 
gives  promise  of  yielding  interesting  results. 

In  the  light  of  these  recent  discoveries  it  is  evident,  that  many  of 
the  physical  constants  in  common  use  may  have  to  be  redeter- 
mined,  but  it  will  probably  be  some  time  before  any  radical  re- 
vision of  the  constants  of  chemical  combination  will  become  neces- 
sary. 

REFERENCES 

Molecular  Physics.     J.  A.  Crowther. 

Conduction  of  Electricity  through  Gases.     Sir  J.  J.  Thomson. 

Rays  of  Positive  Electricity  and  their  Application  to  Chemical  Analysis. 

Sir  J.  J.  Thomson. 
Isotopes.     F.  W.  Aston. 

*Harkins,  Jour.  Am.  Chem.  Soc.,  42,  1328  (1920);   44,  37  (1922). 
fLindemann,  Phil.  Mag.  37,  523  (1919);  38,  173  (1919). 
{Bronsted  and  Hevesy,  Phil.  Mag.  43,  31  (1922);    Zeit.  phys.  Chem. 
90,  189  (1921). 


CHAPTER   XXII 
RADIOACTIVITY 

Discovery  of  Radioactivity.  The  first  radioactive  substance 
was  discovered  by  Henri  Becquerel,*  in  1896.  It  had  been  shown 
by  Roentgen  in  the  previous  year,  that  the  bombardment  of  the 
walls  of  a  vacuum  tube  by  the  cathode  stream,  gives  rise  to  a  new 
type  of  rays,  which,  because  of  their  puzzling  characteristics,  he 
called  X-rays.  The  portion  of  the  tube  where  these  rays  originate 
was  observed  to  fluoresce  brilliantly,  and  it  was  at  once  assumed 
that  this  fluorescence  might  be  the  cause  of  the  new  type  of  radia- 
tion. 

Many  substances  were  known  to  fluoresce  under  the  stimulus  of 
the  sun's  rays,  and  it  was  natural,  in  the  light  of  Roentgen's  dis- 
covery, that  all  substances  which  exhibit  fluorescence  should  be 
subjected  to  careful  examination.  Among  those  who  became 
interested  in  these  phenomena  was  Becquerel.  He  studied  the 
action  of  a  number  of  fluorescent  substances,  among  which  was 
the  double  sulphate  of  potassium  and  uranium.  This  salt,  after 
exposure  to  sunlight,  was  found  to  emit  a  radiation  capable  of 
affecting  a  carefully  protected  photographic  plate.  Further  in- 
vestigation proved,  that  the  fluorescence  had  nothing  to  do  with 
the  photographic  action,  since  both  uranous  and  uranic  salts  were 
found  to  exert  similar  photographic  action,  notwithstanding  the 
fact,  that  uranous  salts  are  not  fluorescent.  The  photographic 
activity  of  both  uranous  and  uranic  salts  was  found  to  be  propor- 
tional to  their  content  of  uranium.  Becquerel  also  showed  that 
preliminary  stimulation  by  sunlight  was  wholly  unnecessary. 
Uranium  salts  which  had  been  kept  in  the  dark  for  years  were 
found  to  be  just  as  active  as  those  which  had  been  recently  ex- 
posed to  brilliant  sunlight 

The  properties  of  the  rays  emitted   by  uranium   salts   differ 
in   many    respects   from    those    of   the    X-rays.     The    rate    of 
emission    of    the    radiation    from    uranium    remains    unaltered 
*  Compt.  rendus,  122,  420  (1896). 
569 


570  THEORETICAL  CHEMISTRY 

at  the  highest  or  the  lowest  obtainable  temperatures.  The 
entire  behavior  of  these  salts  justifies  the  conclusion,  that  the  con- 
tinuous emission  of  penetrating  rays  is  a  specific  property  of  the 
element  uranium  itself.  This  property  of  spontaneously  emitting 
radiations  capable  of  penetrating  substances  opaque  to  ordinary 
light  is  called  radioactivity. 

Discovery  of  Radium.  Shortly  after  the  discovery  of  the 
radioactivity  of  uranium,  the  element  thorium  and  its  compounds 
were  also  found  to  be  radioactive.  As  a  result  of  a  systematic 
examination  by  Mme.  Curie,*  of  minerals  known  to  contain 
uranium  or  thorium,  it  was  learned  that  many  of  these  were  much 
more  radioactive  than  either  uranium  or  thorium  alone.  Thus, 
pitchblende,  one  of  the  principal  ores  of  uranium,  was  found  to 
be  fourtimes  more  active  than  uranium  alone,  while  chalcolite,  a 
phosphate  of  copper  and  uranium,  was  found  to  be  at  least  twice 
as  active  as  uranium.  On  the  other  hand,  when  a  specimen  of 
artificial  chalcolite,  prepared  in  the  laboratory  from  pure  materials, 
was  examined,  its  activity  was  found  to  be  proportional  to  the 
content  of  uranium.  Mme.  Curie  concluded  from  this  result,  that 
natural  chalcolite  and  pitchblende  must  contain  a  minute  amount 
of  some  substance  much  more  active  than  uranium. 

With  the  assistance  of  her  husband,  Mme.  Curie  undertook  the 
task  of  separating  this  unknown  substance  from  pitchblende. 
Pitchblende  is  an  extremely  complex  mineral  and  its  systematic 
chemical  analysis  calls  for  skill  and  patience  of  a  high  order. 
Without  entering  into  details  as  to  the  analytical  procedure,  it 
must  suffice  here  to  state  the  results  obtained.  Associated  with 
the  bismuth  occurring  in  pitchblende,  a  very  active  substance  was 
discovered,  to  which  Mme.  Curie  gave  the  name  polonium  in  honor 
of  her  native  land,  Poland.  In  like  manner,  an  extremely  active 
substance  was  found  associated  with  barium  in  the  alkaline  earth 
group.  This  substance  was  called  radium  because  of  its  great 
radioactivity. 

While  the  isolation  of  pure  polonium  is  extremely  difficult  and, 
while  sufficient  quantities  have  not  been  obtained  to  permit  de- 
terminations of  its  physical  properties,  the  isolation  of  radium 
in  relatively  large  amounts  is  readily  accomplished  The  pure 
bromides  of  radium  and  barium  are  prepared  together,  and  the 
two  salts  are  then  separated  by  a  series  of  fractional  crystalliza- 
*  Compt.  rendus,  126,  1101  (1898). 


RADIOACTIVITY  571 

tions.  That  the  salts  of  barium  and  radium  are  very  similar  in 
chemical  properties,  is  shown  by  the  fact,  that  they  separate  to- 
gether from  the  same  solution.  The  atomic  weight  of  radium  has 
been  determined  by  several  investigators,  the  accepted  value 
being  226.  With  the  exception  of  uranium,  radium  is  the  heaviest 
element  known. 

In  1910,  Mme.  Curie*  succeeded  in  obtaining  metallic  radium. 
It  is  a  metal  possessing  a  silvery  luster,  which  dissolves  in  water 
with  energetic  evolution  of  hydrogen  and  tarnishes  rapidly  in  air 
with  the  formation  of  the  nitride. 

Analysis  has  revealed  the  fact  that  in  all  of  the  principal  uran- 
ium ores,  the  amounts  of  uranium  and  radium  bear  to  each  other 
a  fixed  ratio,  one  part  of  the  latter  being  found  associated  with 
every  three  million  two  hundred  thousand  parts  of  the  former. 
Since  the  amount  of  uranium  occurring  in  pitchblende  may  vary 
from  1  to  50  per  cent,  it  follows  that,  even  with  high  grade  ores, 
several  tons  would  be  required  to  furnish  one  gram  of  pure 
radium. 

lonization  of  Gases.  The  radiations  emitted  by  radioactive 
substances,  like  the  X-rays  have  the  power  of  rendering  the  air 
through  which  they  pass  conductors  of  electricity.  To  ac- 
count for  this  action,  Thomson  and  Rutherford  formulated  the 
theory  of  gaseous  ionization.  According  to  this  theory,  which 
has  since  been  experimentally  confirmed,  the  radiations  break 
up  the  components  of  the  gas  into  positive  and  negative  carriers 
of  electricity  called  ions. 

If  two  parallel  metal  plates  are  connected  to  the  terminals 
of  a  battery,  and  a  radioactive  substance  is  placed  between 
them,  the  air  will  be  ionized  and,  owing  to  the  movement 
of  the  positive  and  negative  ions  toward  the  plates  of 
opposite  sign,  an  electric  current  will  pass  between  the  plates. 
If  the  electric  field  is  weak,  the  mutual  attraction  between  the 
positive  and  negative  ions  will  cause  many  of  them  to  recombine 
before  reaching  the  plates,  and  the  resulting  current  will  be  small. 
As  the  strength  of  the  field  is  increased,  the  greater  will  be  the 
speed  of  the  ions  toward  the  plates  and  the  smaller  will  become 
the  tendency  toward  recombination.  Ultimately,  with  increas- 
ing strength  of  field  all  of  the  ions  will  be  swept  to  the  plates  as 
fast  as  they  are  formed,  and  the  ionization  current  will  attain  a 
*  Compt.  rendus,  151,  523  (1910). 


572  THEORETICAL  CHEMISTRY 

maximum  value.  This  limiting  or,  saturation  current  affords  the 
most  accurate  method  for  the  measurement  of  radioactivity. 

The  method  is  so  sensitive  that,  by  means  of  it  alone,  it  is  pos- 
sible to  detect  amounts  of  radioactive  products  far  beyond  the 
reach  of  the  balance,  or  the  spectroscope.  It  has  been  found,  that 
0.00000002  mg.  of  radium  can'  be  detected  with  certainty. 

The  theory  of  gaseous  ionization  has  been  confirmed  in  several 
different  ways,  but  one  of  the  most  striking  verifications  of  the 
theory  is  that  due  to  C.  T.  R.  Wilson.  Making  use  of  the  fact,  that 
the  ions  tend  to  condense  water  vapor  around  themselves  as  nuclei, 
Wilson  has  succeeded  in  actually  photographing  the  path  of  an 
ionizing  ray  in  air. 

Photographic  Action  of  Radiations.  It  has  already  been 
pointed  out,  that  the  radiations  from  radioactive  substances  are 
capable  of  affecting  a  photographic  plate.  The  photographic 
action  of  the  radiations  has  been  employed  quite  extensively  in 
studying  radioactive  phenomena  from  a  purely  qualitative  stand- 
point. The  method  employed,  consists  in  exposing  the  photo- 
graphic plate,  which  has  previously  been  wrapped  in  opaque  black 
paper,  to  the  action  of  the  radiations.  The  time  of  exposure 
varies  with  the  nature  of  the  substance  under  examination,  a  few 
minutes  being  required  for  highly  active  preparations,  while  sev- 
eral days,  or  even  weeks,  may  be  needed  for  preparations  of  low 
activity. 

Phosphorescence  Induced  by  Radiations.  A  screen  covered 
with  crystals  of  phosphorescent  zinc  sulphide  is  rendered  luminous, 
when  exposed  to  fairly  intense  radiation  from  a  radioactive 
substance.  This  phenomenon  has  been  shown  to  be  due  to  the 
bombardment  of  the  crystals  of  zinc  sulphide  by  the  so-called  a- 
rays  (see  below).  When  the  screen  is  examined  with  a  lens,  the 
phosphorescence  is  seen  to  consist  of  a  series  of  scintillations  of 
very  short  duration. 

Nature  of  the  Radiations.  The  ionizing,  photographic  and 
luminescent  effects  produced  by  the  radiations  from  radioactive 
substances  are  not  sufficient  to  differentiate  them  from  cathode 
rays,  or  from  X-rays,  although  each  of  these  effects  may  be  em- 
ployed to  determine  the  intensity  of  the  radiations. 

Evidence  as  to  the  composite  character  of  the  radiations  was 
furnished  by  a  study  of  their  penetrating  power,  as  well  as  by 
investigations  of  their  behavior,  when  subjected  to  the  action  of 


RADIOACTIVITY  573 

magnetic  or  electric  fields.  A  thin  sheet  of  aluminum,  ur  a  few  centi- 
meters of  air,  was  found  sufficient  to  cut  off  a  large  percentage  of  the 
rays.  The  unabsorbed  portion  of  the  radiation  was  found  to  consist 
of  two  distinct  types,  one  of  which  was  cut  off  by  five  or  six  milli- 
meters of  lead,  while  the  other  possessed  such  great  penetrating 
power  that  its  presence  could  be  readily  detected  after  passing 
through  a  layer  of  lead  fifteen  centimeters  thick. 

Rutherford  named  these  three  distinct  types  of  radiation,  the 
a-,  /?-,  and  7-rays,  respectively.  The  penetrating  powers  of  the 
a-}  fi-}  and  7-rays  may  be  approximately  expressed  by  the  propor- 
tion 1: 100: 10,000;  that  is,  the  /3-rays  are  100  times  more  pene- 
trating than  the  a-rays,  while  the  7-rays  are  100  times  more  pene- 
trating than  the  j3-rays. 

The  general  characteristics  of  the  three  kinds  of  rays  may  be 
briefly  summarized  as  follows :  — 

(1)  a-Rays.     The  a-rays  consist  of  positively  charged  particles 
moving  with  speeds  approximately  one-tenth  as  great  as  that 
of  light.     These  particles  have  been  shown  to  be  identical  with 
helium  atoms  carrying  two  positive  charges  of  electricity.   They 
are  appreciably  deflected  from  a  rectilinear  path  by  magnetic  and 
electric  fields.     They  possess  great  ionizing  power,  but  relatively 
little  penetrating  power,  or  photographic  action.     The  depth  to 
which  an  a-particle  penetrates  a  homogeneous  absorbing  medium 
before  losing  its  ionizing  power,  is  known  as  its  "  range."     The 
range  has  been  found  to  be  proportional  to  the  cube  of  the  initial 
speed  of  the  a-particle,  and  is  one  of  the  characteristic  properties 
of  the  radio-elements  emitting  a-rays. 

(2)  (3-Rays.     The  /3-rays  consist  of  negatively  charged  particles 
moving  with  speeds  varying  from  two-fifths  to  nine-tenths  of  the 
speed  of  light.     They  are,  in  fact,  electrons  moving  with  much 
greater  speeds  than  those  shot  out  from  the  cathode  in  a  vacuum 
tube.     While  the  a-particles  emitted  by  a  particular  radio-ele- 
ment have  a  definite  velocity,  the  corresponding  /3-ray  emission 
consists  of  a  flight  of  particles  having  widely  different  speeds. 
The  penetrating  power  of  the  0-rays  is  conditioned  by  the  speed 
of  the  particles,  those  which  move  most  rapidly  possessing  the 
greatest  penetrating  power.     The  ionizing  action  of  the  0-rays  is 
much  weaker  than  that  of  the  a-rays,  while  exactly  the  reverse  is 
true  of  their  photographic  action. 

(3)  y-Rays.    The    7-rays   are    identical    with   X-rays.     They 


574  THEORETICAL  CHEMISTRY 

consist  of  extremely  short  waves  of  light,  the  wave-length  varying 
from  about  1  X  10~8  cm.,  for  the  rays  of  low  penetrating  power,  to 
about  1  X  10~9  cm.,  for  the  most  penetrating  rays.  Obviously, 
7-rays  cannot  be  deflected  from  a  rectilinear  path  by  either  electric 
or  magnetic  fields. 

The  Radium  Emanation  (Niton).  When  a  specimen  of  radium 
bromide  is  gently  heated,  or  when  a  current  of  air  is  bubbled 
through  a  solution  of  a  radium  salt,  a  small  volume  of  an  exceed- 
ingly radioactive  gas  is  obtained.  Rutherford,  to  whom  we  are 
indebted  for  much  of  our  knowledge  of  this  interesting  substance, 
named  it,  somewhat  vaguely,  the  "  emanation."  In  1911,  Ram- 
say and  Gray,*  making  use  of  a  special  quartz  micro-balance, 
sensitive  to  one  millionth  of  a  milligram,  succeeded  in  determin- 
ing the  mass  of  a  known  volume  of  pure  emanation.  On  the 
assumption  that  the  gas  is  monatomic,  they  computed  the  aver- 
age value  of  its  atomic  weight  to  be  222.  The  value  of  the  atomic 
weight  of  the  emanation,  taken  together  with  its  chemical  inert- 
ness, served  to  determine  its  position  in  the  group  of  non-valent 
elements  in  the  periodic  table.  In  order  to  emphasize  its  rela- 
tionship to  the  inert  gases  Ramsay  proposed  that  it  be  named 
"  niton." 

The  following  crucial  experiment,  carried  out  by  Rutherford 
and  Royds,t  with  niton,  furnished  conclusive  evidence  that  the 
a-particle  consists  of  an  atom  of  helium  carrying  two  positive 
charges.  A  glass  bulb  was  blown  with  walls  thin  enough  to  permit 
the  passage  of  the  a-particles,  but  of  sufficient  strength  to  with- 
stand atmospheric  pressure.  The  bulb  was  filled  with  niton  under 
pressure,  and  then  enclosed  in  an  outer  glass  tube,  to  which  a  spec- 
trum tube  had  been  sealed.  On  exhausting  the  outer  tube,  and 
examining  the  spectrum  of  the  residual  gas,  no  evidence  of  helium 
was  obtained  until  after  an  interval  of  twenty-four  hours.  After 
four  days,  the  yellow  and  green  lines  characteristic  of  helium  were 
plainly  visible,  while  at  the  end  of  the  sixth  day,  the  complete  spec- 
trum of  the  element  was  obtained.  The  unavoidable  conclusion 
from  this  experiment  is,  that  the  presence  of  helium  in  the  outer 
tube  must  have  been  due  to  the  a-particles  which  were  projected 
through  the  thin  walls  of  the  inner  tube.  In  another  experiment, 
the  inrier  tube  was  filled  with  pure  helium  under  pressure,  while  the 

*  Proc.  Roy.  Soc.  A  84,  536  (1911). 
t  Phil.  Mag.,  VI,  17,  281  (1909). 


RADIOACTIVITY  575 

exhausted  outer  tube  was  examined  for  helium.  No  trace  of 
the  gas  could  be  detected  spectroscopically,  even  after  an  interval 
of  several  days,  thus  proving  that  the  helium  detected  in  the  first 
experiment  must  have  resulted  from  the  a-particles  which  had 
been  shot  out  from  the  niton  with  sufficient  energy  to  penetrate 
the  thin  walls  of  the  inner  tube.  These  experiments  leave  no 
room  for  doubt,  that  an  a-particle  becomes  a  helium  atom  when 
its  positive  charge  is  neutralized. 

There  is  an  abundance  of  experimental  evidence  which  proves, 
that  only  one  ex-particle  is  expelled  from  each  atom  of  radium  in 
the  formation  of  niton.  The  process  may  be  represented  by  the 
equation, 

Ra  -  He  =  Nt. 

Since  the  atomic  weight  of  radium  is  226,  and  that  of  helium  is  4, 
it  follows,  that  the  atomic  weight  of  niton  should  be  222.  This,  it 
will  be  remembered,  is  exactly  the  value  which  Ramsay  and  Gray 
obtained  for  the  atomic  weight  of  niton. 

Taken  mass  for  mass,  niton  is  much  more  radioactive  than  the 
radium  from  which  it  was  derived,  but  the  activity  is  found  to 
diminish  rapidly  with  time,  and  practically  to  have  disappeared 
at  the  end  of  a  month. 

A  radium  salt  loses  the  greater  part  of  its  activity  on  de-eman- 
ating, but  if  it  is  examined  from  day  to  day,  it  will  be  found  to  be 
recovering  its  activity  at  precisely  the  same  rate  as  the  niton, 
which  was  derived  from  it,  loses  its  activity.  In  other  words, 
when  the  processes  of  decay  and  recovery  are  examined  quan- 
titatively, it  is  found  that  they  are  complementary,  and  that  the 
total  radioactivity  remains  unaltered.  The  two  processes  of 
decay  and  recovery  follow  an  exponential  law  which  is  expressed 
by  the  following  equations  : 

It  =  /Oe~xr,  (decay) 

and 

It  =  IQ  (1  —  e~xO;    (recovery) 


where  70  is  the  initial  activity,  It  the  activity  after  a  time  tt  \  a 
constant,  known  as  the  radioactive  constant,  and  e  the  base  of  the 
natural  system  of  logarithms.  The  decay  curve  of  niton,  and  the 
recovery  curve  of  the  activity  of  radium,  as  measured  by  the 
a-rays,  are  shown  in  Fig.  139.  From  these  curves,  it  will  be  ap- 


576 


THEORETICAL  CHEMISTRY 


parent  that  while  the  niton  is  decaying  from  day  to  day,  the  ra- 
dium is  in  turn  producing  a  fresh  supply.  This  behavior  is  typical 
of  all  radioactive  bodies  and  illustrates  the  law  of  the  "  conserva- 
tion of  radioactivity." 


40 


2  4  6  8  10  12  14  16 

Days 

Fig.  139 

Counting  the  a-Particles.  Rutherford  and  Geiger  *  devised 
an  electrical  method  for  counting  the  a-particles.  In  their  exper- 
iment the  source  of  the  a-particles  was  a  small  disc  which  had  been 
exposed  to  the  radium  emanation  for  some  hours.  This  disc  was 
placed  in  an  .evacuated  tube,  at  a  measured  distance  from  a  small 
aperture  of  known  cross-section.  The  aperture  was  closed 
with  a  thin  plate  of  mica  through  which  the  a-particles  could 
pass  with  ease.  After  passing  through  the  mica  plate,  the  a- 
particles  entered  an  ionization  chamber  filled  with  air  at  reduced 
pressure,  and  fitted  with  two  charged  metal  plates  connected  with 
appropriate  apparatus  for  measuring  ionization  currents.  When- 
ever an  a-particle  entered  the  ionization  chamber,  a  momentary 
current  passed,  producing  a  sudden  deflection  of  the  needle  of  the 
electrometer.  By  counting  the  number  of  throws  of  the  needle 
occurring  in  a  definite  interval  of  time,  the  total  number  of  a-par- 
ticles  passing  through  the  ionization  chamber  could  be  determined. 
Knowing  the  distance  of  the  source  of  the  radiations  from  the 

*  Proc.  Roy.  Soc.  A,  81,  141  (1908). 


RADIOACTIVITY  577 

% 

aperture,  together  with  the  area  of  the  aperture,  the  total  num- 
ber of  a-particles  emitted  by  the  radioactive  disc,  in  a  given  time, 
could  be  computed.  Rutherford  and  Geiger  thus  found,  that  1 
gram  of  radium  emits  very  nearly  3.4  X  1010  a-particles  per  sec- 
ond. Having  determined  the  total  number  of  a-particles  emitted, 
it  only  remained  to  measure  the  total  charge,  in  order  to  calculate 
the  charge  carried  by  a  single  a-particle.  From  a  series  of  very 
consistent  measurements,  the  charge  carried  by  a  single  a-par- 
ticle  was  found  to  be  9.  3X  10~10  electrostatic  units.  Since  the 
fundamental  charge,  e,  has  been  shown  to  be  4.89  X  10~10  electro- 
static units,  it  follows  that  the  a-particle  carries  two  positive 
charges  of  electricity. 

Avogadro's  Constant  from  Radioactive  Data.  The  rate  at 
which  helium  is  produced  by  radium  has  been  very  carefully 
determined  by  Rutherford  and  Boltwood,*  who  found  it  to  be  0.107 
cu.  mm.  per  day  per  gram  of  radium.  Since  it  has  been  estab- 
lished that  one  gram  of  radium  emits  3.4  X  1010  a-particles  per 
second,  it  follows  that  the  value  of  Avogadro's  constant  is  6.16  X 
1023.  (Let  the  student  verify  this.)  It  will  be  seen,  that  this 
result  agrees  well  with  the  value  previously  found  for  this  con- 
stant from  entirely  different  experimental  data. 

The  Disintegration  Theory.  The  processes  involved  in  the 
transformation  of  the  radio-elements  are  very  different  in  char- 
acter from  those  occurring  in  ordinary  chemical  reactions.  Thus, 
it  has  not  been  found  possible  to  alter,  by  any  known  means,  the 
rate  at  which  the  radio-elements  undergo  change.  Whatever  hy- 
pothesis is  advanced  to  account  for  the  phenomena  of  radioactivity, 
it  must  offer  not  only  a  satisfactory  explanation  of  the  production 
of  a  series  of  active  elements,  differing  from  each  other  and  from 
the  parent  element,  in  both  physical  and  chemical  properties,  but 
also  it  must  account  for  the  enormous  development  of  energy 
which  invariably  accompanies  these  transformations. 

The  theory  of  "  atomic  disintegration,"  advanced  by  Ruther- 
ford and  Soddy,  in  1903,  has  shown  itself  capable,  not  only  of  in- 
terpreting the  extremely  complex  known  facts  of  radioactivity, 
but  also  of  predicting  and  explaining  many  new  ones.  Accord- 
ing to  this  theory,  the  atoms  of  the  radio-elements  are  assumed  to 
undergo  spontaneous  disintegration,  each  atom  passing  through 
a  series  of  well-defined  changes  accompanied,  in  most  cases,  by 
*  Phil.  Mag.  22,  586  (1911). 


578  THEORETICAL  CHEMISTRY 

the  emission  of  a-particles.*  It  is  supposed  that,  on  the  average, 
a  definite  proportion  of  the  atoms  of  each  radio-element  becomes 
unstable  at  a  given  time  and,  as  a  result  of  this  instability,  an 
a-particle  is  expelled  with  great  velocity.  The  expulsion  of  an 
a-particle  leaves  the  new  system  four  units  lighter  than  the  orig- 
inal one,  and  possessing  quite  different  physical  and  chemical 
properties.  This  new  system,  in  turn,  becomes  unstable  and 
expels  another  a-particle,  with  the  simultaneous  production  of  a 
still  lighter  system.  The  process  of  disintegration,  once  begun, 
proceeds  successively  from  one  radio-element  to  the  next,  each 
transformation  taking  place  at  a  definite,  measurable  rate,  until 
ultimately,  a  stable  system  is  produced  and  further  disintegration 
ceases. 

It  has  already  been  pointed  out,  that  a  radio-element  decays 
exponentially  with  time  according  to  the  equation: 

It  =   /06-X',  (1) 

In  this  equation,  the  radioactive  constant  represents  the  frac- 
tion of  the'  total  amount  of  radioactive  substance  undergoing 
disintegration  in  a  unit  of  time,  provided  the  latter  is  so  small 
that  the  quantity  at  the  end  of  the  time  unit  is  only  slightly  differ- 
ent from  the  initial  quantity.  The  reciprocal  of  the  radioactive 
constant  is  called  the  average  life  of  the  element.  Soddy  defines 
the  average  life  of  a  radio-element  as  "  the  sum  of  the  separate 
periods  of  future  existence  of  all  the  individual  atoms,  divided 
by  the  number  in  existence  at  the  starting  point/'  If  nt  repre- 
sents the  number  of  atoms  of  a  radio-element  changing  in  unit 
time,  at  the  end  of  a  time  t,  and  n0)  the  corresponding  value 
when  t  =  0,  equation  (1)  may  be  written, 

nt  =  nQe~xt. 

In  order  to  determine  the  initial  rate  of  change,  let  N0  denote  the 
total  number  of  atoms  originally  present,  and  Nt  the  number  re- 
maining unchanged  at  time  t',  then  we  have, 


/c 


ntdt  =—  e-xr. 
»/* 

But  when  *  =  0,  N0  =  Nt, 
and  A 


Hence  Nt  = 

*  The  emission  of  a  /3-particle  produces  no  appreciable  change  in  weight. 


RADIOACTIVITY  579 

On  differentiating,  we  have 

(2) 


Or,  stated  in  words,  the  rate  at  which  the  atoms  of  a  radio-element 
undergo  disintegration  at  any  given  time  is  proportional  to  the 
total  number  in  existence  at  that  time. 

This  law  of  radioactive  change  is  also  peculiar  to  unimolecular 
chemical  reactions  (see  p.  365).  The  velocity  of  a  unimolecular 
reaction,  however,  is  conditioned  by  the  temperature,  whereas 
the  velocity  of  a  radioactive  change  remains  unaltered  at  the 
highest  and  the  lowest  attainable  temperatures. 

The  time  required  for  one-half  of  a  radio-element  to  undergo 
transformation  is  known  as  the  period  of  half  change,  T,  and  may 
be  readily  calculated  from  X,  as  follows, 
log  2  =  0.4343  XT7 

m      0.6932 
T       — 

Radioactive  Equilibrium.  It  is  evident,  that  a  state  of  equi- 
librium must  ultimately  be  attained  among  the  atoms  of  a  radio- 
active substance.  When  the  rate  of  production  of  a  radio-ele- 
ment from  its  parent  element  is  equal  to  its  rate  of  disintegration 
into  the  next  succeeding  element  of  the  series,  the  substance  is 
said  to  be  in  radioactive  equilibrium.  The  principle  of  radio- 
active equilibrium  may  be  illustrated  by  the  disintegration  of 
radium  into  niton.  Owing  to  the  rapidity  with  which  niton 
disintegrates,  it  does  not  accumulate  continuously,  but  reaches  a 
definite  equilibrium  ratio  with  respect  to  its  parent,  radium,  in 
which  the  amount  of  niton  disappearing  is  just  counter-balanced 
by  the  amount  of  fresh  niton  formed.  If  Xi  is  the  radioactive 
constant  of  radium,  and  N\  is  the  total  number  of  radium  atoms 
involved,  then  the  number  of  radium  atoms  undergoing  disinte- 
gration into  niton  per  second  is  \iNi.  But  when  equilibrium  is 
attained,  \\N\  must  be  equal  to  the  number  of  niton  atoms  dis- 
appearing. If  X2  is  the  radioactive  constant  of  niton,  and  the 
number  of  atoms  of  niton  present  during  equilibrium  is  N2,  the 
number  of  atoms  of  niton  disintegrating  per  second  is  X2JV2. 
Therefore,  we  have 


,Q. 

Xi/X2. 


580  THEORETICAL  CHEMISTRY 

This  mathematical  expression  of  the  jnost  important  law  of  radio- 
active change,  may  be  translated  into  words  as  follows :  —  In 
successive  disintegrations,  each  product  accumulates  until  a  fixed 
ratio  with  respect  to  the  parent  element  is  attained,  and  this  ratio 
is  inversely  proportional  to  the  respective  radioactive  constants  of 
the  two  elements,  or,  is  directly  proportional  to  their  respective 
average  lives.  In  order  that  this  law  may  hold  rigidly,  it  is 
necessary  that  the  period  of  the  parent  element  should  be 
appreciably  longer  than  the  periods  of  any  of  its  products. 
Where  this  is  the  case,  the  product  chosen  need  not  be  the  next 
in  succession,  but  may  be  any  one  of  the  successive  elements  in 
the  series. 

It  is  evident,  that  the  amount  of  niton  in  equilibrium  with  one 
gram  of  pure  radium  is  a  definite  quantity.  At  the  Radiology 
Congress  held  in  Brussels,  in  1910,  it  was  decided  that  this  equi- 
librium quantity  of  niton  should  be  called  the  "  curie"  in  honor 
of  M.  and  Mme.  Curie.  The  volume  of  niton  from  one  gram  of 
radium  can  be  calculated  from  a  knowledge  of  the  number  of 
a-particles  expelled  from  one  gram  of  radium  in  one  second.  It 
has  already  been  stated,  that  this  number  is  3.4  X  1010.  Since 
each  radium  atom,  after  the  expulsion  of  an  a-particle,  becomes 
an  atom  of  niton,  it  follows  that  the  number  of  atoms  of  niton, 
q,  formed  per  gram  per  second  is  3.4  X  1010.  When  equilibrium  is 
attained,  the  number  of  atoms  of  niton,  N,  present  per  gram  will 
be  N  =  9/X.  Since  X  =  2.085  X  10~6  per  second,  we  have  N  = 
1.63  X  1017.  Since  there  are  2.75  X  1019  molecules  present  in 
one  cubic  centimeter  of  any  gas,  under  standard  conditions  of 
temperature  and  pressure,  the  volume  of  one  curie  of  niton  will 
be  0.59  cubic  millimeter.  Rutherford  found,  by  direct  experiment, 
0.6  cubic  millimeter  as  the  volume  of  one  curie  of  nitron,  which 
is  in  remarkably  close  agreement  with  the  calculated  value.  Since 
a  curie  is  a  relatively  large  unit  in  comparison  with  the  amounts 
of  niton  ordinarily  obtainable,  it  is  convenient  to  use  as  a 
practical  unit,  the  milli-curie,  which  is,  as  its  name  implies,  the 
amount  of  niton  in  equilibrium  with  ten  milligrams  of  pure 
radium. 

The  Origin  of  Radium  and  the  Uranium  Disintegration  Series. 
As  the  result  of  a  large  amount  of  careful  experimental  work,  it 
has  been  shown,  that  radium  is  but  one  of  a  series  of  radio-elements, 
each  of  which  is  formed  by  the  atomic  disintegration  of  the  pre- 


RADIOACTIVITY  581 

ceding  member  of  the  series,  in  precisely  the  same  manner  as  niton 
is  formed  from  radium. 

Very  soon  after  the  discovery  of  radium,  it  was  suspected  that 
some  intimate  connection  might  be  found  to  exist  between  radium 
and  uranium,  but  it  was  not  until  after  Strutt  and  Boltwood, 
from  an  examination  of  a  large  number  of  uranium-bearing  min- 
erals, had  established  the  existence  of  a  constant  ratio  between  the 
percentages  of  uranium  and  radium,  that  this  suspicion  was  con- 
firmed. 

Radium  is  not  the  direct  product  of  uranium,  but  rather 
is  the  fifth  successive  member  of  the  disintegration  series  which 
has  its  origin  in  uranium.  Uranium  changes  into  UXi  with  the 
simultaneous  expulsion  of  an  a-particle  and  a  consequent  loss  of 
four  units  in  atomic  weight.  The  element  uranium  disintegrates 
at  an  exceedingly  slow  rate,  the  probable  value  of  its  average  life 
being  8,000,000,000  years.  The  element  UXi  changes  with  the 
emission  of  a  0-particle,  and  hence  without  loss  of  weight,  into 
the  element  UX2  which,  because  of  its  brief  average  life  of  1.65 
minutes,  is  sometimes  called  brevium.  Two  other  radio-elements, 
UII  and  lo,  with  periods  of  average  life  of  3,000,000  and  100,000 
years  respectively,  are  formed  in  succession  from  UX2  before  the 
element  radium  is  formed  from  ionium.  Since  three  a-particles 
are  expelled  in  the  course  of  the  five  successive  disintegrations 
involved  in  the  production  of  radium  from  uranium,  it  follows, 
that  the  atomic  weight  of  radium  should  be  3X4=  12  units 
lighter  than  the  atomic  weight  of  uranium,  or  238  —  12  =  226. 
It  will  be  seen,  that  the  experimentally  determined  value  of  the 
atomic  weight  of  radium  agrees  perfectly  with  that  predicted  from 
the  disintegration  theory. 

If  a  negatively  charged  wire  be  introduced  into  a  vessel  con- 
taining niton,  an  extremely  radioactive  substance,  known  as  the 
"  active  deposit,"  is  condensed  upon  its  surface.  This  deposit, 
which  can  be  detected  by  neither  balance  nor  microscope,  has 
been  found  to  consist  of  the  successive  disintegration  products 
of  niton.  The  atomic  disintegration  of  niton  takes  place  in  eight 
distinct  successive  stages,  the  first  four  radio-elements  of  the  active 
deposit  being  characterized  by  very  short  average  lives,  ranging 
from  38.5  minutes  for  RaB,  to  one-millionth  of  a  second  for 
RaC',  while  the  remaining  elements  of  the  series  undergo  trans- 
formation at  a  much  slower  rate,  their  average  lives  varying  from 


582  THEORETICAL  CHEMISTRY 

7.25  days  to  24  years.  It  is  of  interest  to  note,  that  RaF  has  been 
found  to  be  identical  with  polonium,  the  first  radio-element  to  be 
isolated  from  pitchblende  by  Mme.  Curie. 

While  lack  of  space  precludes  a  detailed  consideration  of  thor- 
ium and  actinium,  it  should  be  mentioned  that  each  of  these  ele- 
ments disintegrates  in  a  similar  manner  to  uranium,  giving  rise 
to  a  series  of  interesting  radio-elements  which,  in  general,  are 
shorter  lived  than  the  members  of  the  uranium  series.  Taking 
the  three  series  together,  the  science  of  radioactivity  now  embraces 
nearly  forty  elements  each  of  which  is  endowed  with  specific  proper- 
ties and  with  periods  of  existence  varying  from  thousands  of  mil- 
lions of  years  on  the  one  hand,  to  a  small  fraction  of  a  second  on  the 
other.  Soddy  says  of  the  radio-elements,  "  It  is  unlikely  that  any 
more  of  these  unstable  elements  remain  to  be  discovered,  unless 
some  entirely  unknown  and  unsuspected  source  of  radioactive 
materials  is  found."  The  table  shown  on  p.  583  gives  the  three 
disintegration  series  as  arranged  by  Soddy. 

The  numbers  within  the  circles  denote  the  atomic  weights  of 
the  elements,  while  the  small  circles  and  dots  at  the  right  of  the 
larger  circles  indicate  the  character  of  the  radiation  given  out  at 
each  stage  of  the  disintegration  process.  The  average  life,  1/X, 
of  each  element  in  the  series  is  given  below  the  name  of  the  ele- 
ment. 

The  Ultimate  Product  of  Radioactive  Change.  It  will  be  ob- 
served that  the  end-product  of  the  uranium  series  is  an  inactive 
element,  having  an  estimated  atomic  weight  of  206.  If  this  is 
really  an  inactive  product,  and  not  an  element,which  is  undergoing 
transformation  at  such  an  excessively  slow  rate  as  to  be  beyond 
the  reach  of  our  present  experimental  methods,  it  follows  that  the 
element  must  have  been  continuously  accumulating  in  all  of  the 
minerals  containing  uranium  since  their  original  formation,  and 
furthermore,  that  it  must  be  one  of  the  more  common  elements. 
Such  an  element  is  lead,  which  not  only  is  found  in  all  of  the  uran- 
ium minerals,  with  the  exception  of  autunite,  but  also  is  present 
in  amounts  which  are  proportional  to  the  age  of  the  geological 
formation  from  which  the  mineral  is  obtained.  There  is  an  abun- 
dance of  indirect  evidence  which  makes  it  appear  highly  probable, 
that  the  ultimate  product  of  the  disintegration  of  uianium  is 
an  isotopic  form  of  lead.  It  is  hoped,  however,  that  direct  evi- 
dence on  this  question  may  soon  be  forthcoming,  since  Mme, 


RADIOACTIVITY 


583 


Curie  and  Dr.  Debierne  have  undertaken  the  separation  of  a 
large  amount  of  polonium  from  pitchblende,  with  a  view  to  prov- 
ing, by  the  aid  of  the  spectroscope,  that  the  disintegration  prod- 
uct of  polonium  is  really  lead. 

TABLE  SHOWING  DISINTEGRATION  SERIES 


584 


THEORETICAL  CHEMISTRY 


The  estimated  atomic  weights  of  the  end-products  of  the  other 
two  series  are  206  or  210,  in  the  case  of  the  actinium  series,  and  208, 
in  the  case  of  the  thorium  series.  These  values  suggest  the  possi- 
bility that  the  ultimate  products  of  these  series  also  may  be  isotopic 
forms  of  lead.  The  evidence,  however,  in  the  case  of  thorium  is  far 
less  definite  than  in  that  of  uranium,  while  in  the  case  of  actinium,  it 
may  be  said,  that  in  the  absence  of  definite  knowledge  of  its  atomic 
weight  or  that  of  any  of  its  disintegration  products,  deduction  of 
the  atomic  weight  of  the  end-product  of  the  series  is  hardly 
justifiable. 

The  atomic  weight  of  lead,  obtained  from  radioactive  minerals 
collected  at  widely  different  points,  has  been  determined  with 
extreme  accuracy  by  Richards  and  his  co-workers.*  As  the  follow- 
ing table  shows,  all  of  the  values  of  the  atomic  weights  are  less 
than  207.20,  the  value  recently  found  by  Baxter  and  Grover,f 
as  a  result  of  their  precise  determinations  of  the  atomic  weight 
of  inactive  lead  obtained  from  various  sources. 


ATOMIC   WEIGHT  OF  LEAD  OF  RADIOACTIVE  ORIGIN 


Source 

Atomic 
Weight 

North  Carolina  uraninite  

206.40 

Joachimsthal  pitchblende  

206.57 

Colorado  carnotite 

206  59 

Ceylon  thorianite 

206  82 

English  pitchblende 

206.86 

Australian  carnotite.  .  . 

206.34 

Norwegian  cleveite  

206.08 

Ordinary  lead  

207.20 

Notwithstanding  the  variation  in  the  values  of  the  atomic  weight 
of  lead  derived  from  radioactive  sources,  the  ultra-violet  spectra 
of  the  different  specimens  have  been  shown  to  be  identical  with  the 
spectrum  of  ordinary  inactive  lead.  About  one  kilogram  of  lead 
from  a  radioactive  source  was  converted  by  Richards  and  Hall  J 
into  the  nitrate,  which  was  then  subjected  to  a  series  of  fractional 

*  Jour.  Am.  Chem.  Soc.  36,  1329  (1914);  38,  1658,  2613  (1916);  40,  1409 
(1918). 

t  Ibid.  37,  1058  (1915). 

t  Jour.  Am.  Chem.  Soc.  39,  531  (1917). 


RADIOACTIVITY  585 

crystallizations  with  a  view  to  effecting  a  possible  separation  of 
isotopic  forms.  Notwithstanding  the  fact,  that  over  one  thousand 
crystallizations  were  performed,  no  indication  was  obtained  of  any 
separation  having  been  brought  about.  Not  only  are  the  solubili- 
ties of  the  isotopic  nitrates  identical,  but  also  their  spectra  and 
refractive  indices  are  likewise  the  same. 

Total  Energy  Evolved  by  the  Complete  Disintegration  of  Ra- 
dium. Curie  and  Laborde  *  were  the  first  to  call  attention  to  the 
interesting  fact,  that  the  temperature  of  radium  compounds  was 
uniformly  higher  than  that  of  their  environment.  Careful  meas- 
urements have  shown,  that  one  gram  of  radium  evolves  heat  at 
the  rate  of  approximately  133  gram-calories  per  hour.  A  knowl- 
edge of  this  constant,  together  with  the  value  of  the  average  life 
of  radium,  enables  us  to  calculate  the  total  number  of  calories 
of  heat  energy  which  will  be  evolved  in  the  course  of  the  com- 
plete disintegration  of  a  given  mass  of  radium.  Thus,  the  energy 
evolved  by  one  gram  of  radium  during  its  complete  disintegration 
is  computed  as  follows:  —  since  there  are  8,760  hours  in  one  year, 
one  gram  of  radium  will  evolve  133  X  8,760  =  1,160,000  calories 
per  year  and,  since  the  average  life  of  radium  is  2,440  years,  the 
total  energy  evolved  in  its  disintegration  will  be,  1,160,000  X 
2,440  =  2,830,400,000  calories.  To  give  some  idea  of  the  signi- 
ficance of  these  figures,  it  may  be  stated  that  this  amount  of  energy 
is  equivalent  to  one-million  times  the  amount  of  heat  developed 
in  the  combustion  of  an  equal  weight  of  high-grade  coal. 

REFERENCES 

The  Interpretation  of  Radium,  4th  Edition.     Soddy. 
Radioactive   Substances   and   their   Radiations.     Rutherford. 
Radioactive  Transformations.     Rutherford, 

*  Compt.  rendus,  136,  673  (1903). 


CHAPTER  XXIII 
ATOMIC   STRUCTURE 

The  Modern  Conception  of  Atomic  Structure.  As  a  result  of 
the  investigations  of  Thomson,*  Rutherford,  f  Nicholson,  {  Bohr,§ 
and  others,  a  theory  of  atomic  structure  has  been  developed,  which 
affords  a  satisfactory  interpretation  of  many  of  the  important 
relationships  among  the  chemical  elements. 

Briefly  stated,  this  theory  assumes  that  the  atom  consists  of  a 
central,  positively  charged  nucleus,  surrounded  by  a  miniature 
solar  system  of  electrons.  The  investigations  of  Rutherford  and 
Geiger||  show,  that  the  character  of 'the  deflection  of  a-particles 
shot  out  from  radioactive  atoms  at  speeds  approximating  20,000 
miles  per  second,  and  consequently  completely  penetrating  other 
atoms,  is  such  as  to  indicate  an  extremely  high  concentration  of 
positive  electricity  on  the  central  nucleus.  The  central  nucleus 
which  is  supposed  to  represent  nearly  the  entire  mass  of  the  atom, 
is  thought  to  be  very  small  in  comparison  with  the  size  of  the  atom 
as  a  whole.  Recent  investigations  make  it  appear  probable,  that 
the  maximum  diameter  of  the  nucleus  of  the  hydrogen  atom  is 
about  one  one-hundred-thousandth  of  the  diameter  commonly 
attributed  to  the  atom.  In  commenting  on  this  statement,  Har- 
kins  says:  —  If  "  On  this  basis  the  atom  would  have  a  volume  a 
million-billion  times  larger  than  that  of  its  nucleus,  and  thus  the 
nucleus  of  the  atom  is  much  smaller  in  comparison  with  the  size 
of  the  atom  than  is  the  sun  when  compared  with  the  dimensions 
of  its  planetary  system."  It  is  highly  probable  that  the  central 
nucleus  is  itself  made  up  of  a  definite  number  of  units  of  positive 
electricity,  together  with  a  small  number  of  attendant  electrons. 

It  is  further  assumed  that  the  units  of  positive  electricity  are 

*  Phil.  Mag.,  7,  237  (1904). 
t  Popular  Science  Monthly,  87,  105  (1915). 
t  Phil.  Mag.,  22,  864  (1911). 
§  Phil.  Mag.,  26,  476,  857  (1913). 
||  Phil.  Mag.,  21,  669  (1911). 
If  Science,  66,  419  (1917). 
586 


ATOMIC  STRUCTURE  587 

hydrogen  atoms,  each  of  which  has  been  deprived  of  one  electron. 
If  the  mass  of  an  atom  is  largely  due  to  the  presence  of  hydrogen 
nuclei,  then  we  should  expect  Prout's  hypothesis  to  hold,  and  the 
atomic  weights  of  the  elements  to  be  exact  multiples  of  the  atomic 
weight  of  hydrogen.  When  we  consider,  however,  that  according 
to  the  electromagnetic  theory,  the  total  mass  of  a  body  composed 
of  positive  and  negative  units  is  dependent  upon  the  relative  posi- 
tions of  these  units  when  packed  together,  it  is  evident,  that  the 
mass  of  the  atom  will  not  necessarily  be  an  exact  multiple  of  the 
mass  of  the  hydrogen  atom. 

It  has  already  been  pointed  out,  that  helium  is  a  product  of 
many  radioactive  transformations.  This  fact  may  be  taken  as  an 
indication  of  the  extraordinary  stability  of  the  helium  atom.  Be- 
cause of  its  stability,  the  nucleus  of  this  atom  has  come  to  be  con- 
sidered as  a  secondary  unit  of  positive  electricity.  The  nucleus 
of  the  helium  atom,  or  the  nucleus  of  an  a-particle  is  assumed  to 
consist  of  four  hydrogen  nuclei  with  two  nuclear  electrons. 

The  Atomic  Number.  Since  the  algebraic  sum  of  the  positive 
and  negative  electrification  on  an  atom  must  be  zero,  it  follows, 
that  the  excess  positive  charges  resident  upon  the  nucleus  must 
be  equal  to  the  number  of  electrons  outside  the  nucleus.  This 
number,  which  has  come  to  be  recognized  as  more  important  and 
characteristic  than  the  atomic  weight,  is  known  as  the  atomic 
number. 

X-Rays  and  Atomic  Structure.  The  discovery  by  W.  L.  Bragg, 
in  1912,  that  X-rays  undergo  reflection  at  crystal  surfaces,  and  the 
subsequent  development  by  Mr.  Bragg  and  his  father,  W.  H. 
Bragg,  the  inventor  of  the  X-ray  spectrometer,  has  led  to  a  series 
of  investigations  of  the  utmost  importance  to  both  the  chemist  and 
the  physicist. 

The  bombardment  of  metal  plates,  usually  of  platinum,  by  elec- 
trons gives  rise  to  X-rays.  The  radiation  issuing  from  an  X-ray 
tube  is  very  far  from  homogeneous.  It  has  been  found,  how- 
ever, that  every  substance,  when  properly  stimulated,  is  ca- 
pable of  emitting  a  homogeneous  and  characteristic  X-radiation, 
the  penetrating  power  of  which  is  wholly  determined  by  the  nature 
of  the  elements  of  which  the  substance  is  composed.  The  pene- 
trating power  of  this  typical  X-radiation  increases  with  the  atomic 
weight  of  the  radiating  element.  With  elements  whose  atomic 
weights  are  less  than  24,  the  radiation  is  too  feeble  to  be  easily 


588  THEORETICAL  CHEMISTRY 

measured.  It  is  important  to  note,  that  this  property  of  the 
elements  is  not  a  periodic  function  of  the  atomic  weight.  This 
type  of  X-radiation  is  entirely  independent  of  external  conditions, 
indicating  that  it  is  closely  connected  with  the  internal  structure 
of  the  atoms  from  which  it  emanates. 

By  making  use  of  the  reflecting  power  of  one  of  the  cleavage 
planes  of  a  crystal,  and  employing  different  metals  as  anti-cath- 
odes in  an  X-ray  tube,  Moseley  *  succeeded  in  photographing 
the  X-ray  spectra,  characteristic  of  a  number  of  the  elements. 
He  showed  that  the  X-ray  spectrum  of  an  element  is  ex- 
tremely simple  and  consists  of  two  groups  of  lines,  known 
as  the  "  K  "  and  "  L  "  radiations.  As  a  result  of  careful  study 
of  the  "  K  "  radiations  of  thirty-nine  elements  from  aluminium 
to  gold,  Moseley  discovered  that  these  radiations  are  charac- 
terized by  two  well-defined  lines  whose  vibration  frequency,  v, 
is  connected  with  the  atomic  number  of  the  element  N,  by  the 
simple  relation, 

,-A(N-  I)2,  (1) 

where  A  is  a  constant. 

It  has  recently  been  shown  by  Duane,f  that,  for  the  elements 
from  cerium  to  magnesium,  the  square  root  of  the  frequency, 
v,  is  not  quite  a  linear  function  of  the  atomic  number,  N.  This 
departure  from  a  straight  line  relation  becomes  apparent  when 
the  square  root  of  v  is  plotted  against  N,  as  in  Fig.  140. 

When  the  elements  are  arranged  in  the  order  of  their  atomic 
numbers  instead  of  in  the  order  of  their  atomic  weights,  the  irregu- 
larities t  hitherto  noted  in  connection  with  argon,  cobalt,  and 
tellurium  entirely  disappear. 

In  reviewing  Moseley's  work  on  X-ray  spectra,  Soddy  §  says:  — 
"  A  veritable  roll-call  of  the  elements  has  been  made  by  this 
method.  Thirty-nine  elements,  with  atomic  weights  between 
those  of  aluminium  and  gold,  have  been  examined  in  this  way,  and 
in  every  case,  the  lines  of  the  X-ray  spectrum  have  been  found  to 
be  simply  connected  with  the  integer  -that  represents  the  place 
assigned  to  it  by  chemists  in  the  periodic  table." 

*  Phil.  Mag.,  26,  210  (1913);  27,  703  (1914). 

t  Phys.  Rev.,  14,  516  (1919). 

J  See  p.  555. 

§  Ann.  Reports  on  the  Prog,  of  Chemistry,  p.  278  (1914). 


ATOMIC  STRUCTURE 


589 


6.0 


4.5 


4.0 


?      3.5 
o 

X 

g  3.0 

3 


2.0 


1.6 


1.0 


10  20  30  40  50  60  70  80 

Atomic  Number 
Fig.  140 

One  of  the  most  interesting  results  of  this  "  roll-call  "  of  the 
elements  is  the  fixing  of  the  number  of  possible  rare-earth  elements. 
Between  barium  and  tantalum  there  are  places  for  only  fifteen 
rare-earth  elements  and  fourteen  of  these  places  are  filled.  While 
future  investigation  may  necessitate  some  rearrangement  in  the 
order  of  tabulation,  the  total  number  of  these  elements  is  limited 
to  fifteen. 

Periodicity  among  the  Radio-elements.  The  problem  of  plac- 
ing the  newly  discovered  radio-elements  in  the  periodic  table  re- 
mained unsolved  until  1913,  when  Fajans  *  and  Soddy,f  working 
independently,  discovered  an  important  generalization  concerning 
the  changes  in  chemical  properties  resulting  from  the  expulsion 
of  a-  and  /3-particles  during  radioactive  transformations.  This 

*  Physikal.  Zeit.,  14,  49  (1913). 
t  Chem.  News,  107,  97  (1913). 


590 


THEORETICAL  CHEMISTRY 


important  generalization  may  be  stated  as  follows :  —  The  expul- 
sion of  an  a-partide  causes  a  radioactive  element  to  shift  its  position 
in  the  periodic  table  two  places  in  the  direction  of  decreasing  atomic 
weight,  whereas  the  emission  of  a  (3-particle  causes  a ,  shift  of  one 
place  in  the  opposite  direction.  This  generalization  not  only  agrees 
with  our  present  theory  of  atomic  structure,  but  may  be  shown 
to  be  a  necessary  consequence  of  this  theory. 

The  loss  of  an  a-particle,  or  helium  atom,  involves  a  loss  of  4 
units  in  atomic  weight  and  of  2  units  of  positive  electricity  from 
\he  nucleus  of  the  atom.  In  consequence  of  this  loss,  the  atomic 


RADIO-ELEMENTS  AND  PERIODIC  LAV/ 
ALL  ELEMENTS  IN  THE  SAME  PLACE 

IN  THE   PERIODIC  TABLE 
ARE  CHEMICALLY  NON-SEPARABLE 
>V  AND   (PROBABLY) 

iui    X^PECTROSCOPICALLY  INDISTINGUISHABLE 


Fig.  141 
*  i 

number  is  diminished  by  2  units,  and  the  resulting  new  element 
will  find  a  place  in  the  periodic  table  two  groups  to  the  left  of  that 
occupied  by  the  parent  element.  On  the  contrary,  while  the 
expulsion  of  a  0-particle,  or  electron,  involves  practically  no 
change  in  mass,  the  nucleus  of  the  parent  atom  suffers  a  loss  of 
1  unit  of  negative  electricity.  This  loss  is  equivalent  to  a  gain  of 
1  unit  of  positive  electricity,  or  to  an  increase  of  1  unit  in  the 
atomic  number,  and  in  consequence,  the  position  of  the  new  element 


ATOMIC   STRUCTURE  591 

in  the  periodic  table  will  be  shifted  one  group  to  the  right  of  that 
occupied  by  the  parent  element,  with  a  corresponding  change  in 
valence. 

Soddy's  arrangement  of  all  of  the  radio-elements  in  accordance 
with  this  generalization  is  shown  in  Fig.  141.  Thus,  starting  with 
the  element  uranium  in  Group  VIA,  we  may  follow  the  successive 
steps  in  the  radium  disintegration  series  which  was  discussed  in 
the  preceding  chapter.  The  element  UXi,  resulting  from  U  by 
the  loss  of  an  a-particle,  is  placed  in  Group  IVA.  This  element  in 
turn  undergoes  a  |3-ray  change,  producing  the  element  UX2,  which 
is  accordingly  placed  in  Group  VA.  The  element  UII  in  Group 
VIA  is  formed  from  UX2  by  /3-ray  disintegration,  while  the 
element  lo  results  from  the  loss  of  an  a-particle  by  UII0 
with  a  consequent  shift  to  the  left  into  Group  IVA.  A 
similar  loss  of  an  a-particle  by  lo  brings  us  to  the  element 
Ra,  in  Group  IIA.  In  the  successive  steps  of  this  disinte- 
gration from  U  to  Ra,  three  a-particles,  or  12  units  of  atomic 
mass,  are  lost  and  the  atomic  weight  of  Ra,  as  calculated  from  that 
of  U,  agrees  with  the  atomic  weight  found  by  direct  experiment. 
In  like  manner,  the  remaining  stages  of  the  disintegration  may  be 
followed  to  the  end-product  in  Group  IVB. 

Isotopes  and  Isobares.  Perhaps  the  most  striking  feature  in 
the  table  is  the  occurrence  of  several  different  elements  in  the 
same  place,  as  for  example  in  Group  IVB,  where  in  the  place  occu- 
pied by  the  element  Pb,  we  also  find  RaB,  RaD,  ThB,  and  AcB, 
together  with  four  other  elements  to  which  no  names  have  been 
assigned,  but  which  are  none  the  less  stable  end-products.  The 
individual  members  of  such  a  group  of  elements  occupying  the 
same  place  in  the  periodic  table,  and  being  in  consequence  chem- 
ically identical,  are  therefore  isotopic. 

Just  as  it  is  possible  to  have  elements  with  identical  chemical 
properties  but  differing  in  atomic  weight,  so  it  is  also  possible  to 
have  elements  possessing  different  chemical  properties  with  the 
same  atomic  weight.  Such  elements  have  been  called  by 
Stewart,*  isobar es.  The  product  of  a  radioactive  change  in 
which  a  /3-particle  is  lost  is  an  isobare  of  the  parent  element,  since 
its  position  in  the  periodic  table  has  been  shifted  without  under- 
going appreciable  change  in  mass,  while  its  chemical  properties 
have  changed  to  correspond  with  its  new  position  in  the  table. 
*  Phil.  Mag.  36,  326  (1918). 


592  THEORETICAL  CHEMISTRY 

It  is  worthy  of  note,  that  no  isofrare  has  yet  been  discovered 
among  the  elements  which  are  not  radioactive. 

It  is  also  interesting  to  note  in  Soddy's  table  (Fig.  144),  that 
"  the  ten  occupied  spaces  (groups)  contain  nearly  forty  distinct 
elements,  whereas  if  chemical  analysis  alone  had  been  available  for 
their  recognition,  only  ten  elements  could  have  been  distinguished. " 

The  Hydrogen-Helium  System  of  Atomic  Structure.  A 
generalization  s  milar  to  that  just  outlined  for  the  radio-elements 
has  been  found  by  Harkins  and  Wilson  *  to  hold  true  for  the 
lighter  elements  which  apparently  do  not  undergo  appreciable 
a-ray  disintegration.  Beginning  with  helium,  and  adding  4  units 
of  atomic  weight  for  each  increase  of  2  units  in  the  atomic  number, 
gives  the  atomic  weights  of  the  elements  in  the  even-numbered 
groups  of  the  periodic  table,  neglecting  small  changes  in  mass  due 
to  nuclear  packing.  This  rule  has  been  found  to  hold  very  closely 
for  all -of  the  elements  having  atomic  weights  below  60. 

The  atomic  weights  of  the  elements  of  the  odd-numbered  groups 
can  be  calculated  by  a  similar  rule,  provided  that  the  atom  of 
lithium,  the  first  member  of  the  odd-numbered  groups,  be  assumed 
to  be  made  up  of  1  hydrogen  and  3  helium  nuclei.  The  following 
table  gives  the  results  as  calculated  by  Harkins  and  Wilson  for  the 
first  three  series  of  the  periodic  table. 

*  Proc.  Nat.  Acad.,  Vol.  I,  p.  276  (1915). 


ATOMIC  STRUCTURE 


593 


00 


o 

I 


+S8 

^    O>  C5  Oi 


oo 


C3  O  CO 


w 

+00 


Q  • 

o 


w 

+00 


•    0)  -U 

lea 


2* 


8  1 

w  2 


594  THEORETICAL  CHEMISTRY 

The  so-called  theoretical  atomic,  weights  are  calculated  on  the 
basis  H  =  1,  while  the  experimentally  determiried  values  are  on 
the  basis  0  =  16,  or  H  =j  1.0078.  iThe  remarkably!  close  agree- 
ment between  the  two  sets  of  values  is  taken  as  an  indication  that 
the  packing  effect,  resulting  from  the  formation  of  the  elements 
from  hydrogen  nuclei  and  attendant  electrons,  is  very  small. 
This  packing  effect  has  been  estimated  to  involve  a  decrease  in 
atomic  mass  of  about  0.77  per  cent,  and  is  believed  to  be  due 
almost  entirely  to  the  formation  of  the  helium  atom.  The  hydro- 
gen-helium hypothesis  of  atomic  structure  offers  a  rational  ex- 
planation of  many  interesting  but  hitherto  obscure  facts  concern- 
ing the  nature  of  the  elements. 

Relation  between  Atomic  Weights  and  Atomic  Numbers.  For 
all  elements  whose  atomic  weights  are  less  than  that  of  nickel, 
Harkins  finds  the  following  simple  mathematical  relation  to  hold, 

W  =  2  JV  +  i+i  (-  I)"-1,  (2) 

where  W  is  the  atomic  weight  and  N  is  the  atomic  number.  In 
other  words,  the  atomic  weights  are  &  linear  function  of  the  atomic 
numbers. 

The  Periodic  Table  of  Harkins.  In  the  light  of  recent  dis- 
coveries the  periodic  law  acquires  new  significance;  in  fact,  to- 
day the  periodic  law  may  be  regarded  as  the  most  comprehensive 
generalization  in  the  whole  science  of  chemistry. 

Attention  has  already  been  directed  in  an  earlier  chapter  to  the 
most  apparent  of  the  imperfections  in  MendeleefFs  system  of 
classification  of  the  elements.  While  the  later  '•  tables  are  more 
complete  than  the  original,  owing  in  Ipart  to  the  discovery  of  new 
elements,  it  must  be  admitted,  nevertheless,  that  relatively  little 
real  progress  has  been  made,  until  recently,  toward  removing  the 
seemingly  inherent  defects  of  the  system. 

A  satisfactory  periodic  table  should  meet  the  following  require- 
ments : 

(1)  It  should  afford  a  place  for  isotopic  elemehts,  such  as  lead. 

(2)  The  radio-elements,  together  with  their  i-  and  ^-disinte- 
gration products,  should  be  shown. 

(3)  It  should  contain  no  vacant  spaces,  except  those  correspond- 
ing to  the  atomic  numbers  of  undiscovered  elements. 

(4)  It  should  bring  out  the  relation  between  the  elements  con- 
stituting a  main  group,  and  those  forming  the  corresponding  sub- 


ATOMIC   STRUCTURE  595 

group.  For  example,  the  relation  between  the  elements  Be,  Mg, 
Ca,  Sr,  Ba,  and  Ra,  on  the  one  hand,  and  the  elements  Zn,  Cd,  and 
Hg,  on  the  other,  should  be  emphasized. 

(5)  The  elements  of  Group  O  and  Group  VIII  should  fit  natu- 
rally in  the  table. 

(6)  All  of  the  foregoing  conditions  should  be  shown  by  means 
of  a  continuous  curve  connecting  the  elements  in  the  order  of  their 
atomic  numbers,  the  latter  having  been  shown  to  be  more  charac- 
teristic of  an  element  than  its  atomic  weight. 

A  table  which  satisfactorily  meets  these  requirements  has  re- 
cently been  devised  by  Harkins  and  Hall.  This  table  may  be  con- 
structed in  the  form  of  a  helix  in  space,  or  as  a  spiral  in  a  plane. 
The  following  description  of  the  helical  arrangement,  shown  in 
Fig.  142,  is  taken  verbatim  from  the  original  paper  of  Harkins  and 
Hall.* 

"  The  atomic  weights  are  plotted  from  the  top  down,  one  unit  of 
atomic  weight  being  represented  by  one  centimeter,  so  the  model 
is  about  two  and  one-half  meters  high.  .  .  . 

"  The  balls  representing  the  elements  are  supposed  to  be  strung 
on  vertical  rods.  All  of  the  elements  on  one  vertical  rod  belong  to 
one  group,  have  on  the  whole  the  same  maximum  valence,  and  are 
represented  by  the  same  color.  The  group  numbers  are  given  at 
the  bottom  of  the  rods.  On  the  outer  cylinder  the  electro-nega- 
tive elements  are  represented  by  black  circles  at  the  back  of  the 
cylinder,  and  electro-positive  elements  by  white  circles  on  the 
front  of  the  cylinder.  The  transition  elements  of  the  zero  and 
fourth  groups  are  represented  by  circles  which  are  half  black  and 
half  white.  The  inner  loop  elements  are  intermediate  in  their 
properties.  Elements  on  the  back  of  the  inner  loop  are  shown 
as  heavily  shaded  circles,  while  those  on  the  front  are  shaded  only 
slightly. 

"  In  order  to  understand  the  table  it  may  be  well  to  take  an 
imaginary  journey  down  the  helix,  beginning  at  the  top.  Hydro- 
gen (atomic  number  and  atomic  weight  =  1)  stands  by  itself,  and 
is  followed  by  the  first  inert,  zero  group,  and  zero  valent  element 
helium.  Here  there  comes  the  extremely  sharp  break  in  chemical 
properties  with  the  change  to  the  strongly  positive,  univalent 
element  lithium,  followed  by  the  somewhat  less  positive  bivalent 
element,  beryllium,  and  the  third  group  element  boron,  with  a 
*  Jour.  Am.  Chem.  Soc.,  38,  169  (1916). 


596 


THEORETICAL  CHEMISTRY 


Or 


20 


4D 


60 


80 


100 


120 


140 


160 


180 


220 


240 


Group 


Fig.  142 


ATOMIC   STRUCTURE  597 

positive  valence  of  three,  and  a  weaker  negative  valence.  At  the 
extreme  right  of  the  outer  cylinder  is  carbon,  the  fourth  group 
transition  element,  with  a  positive  valence  of  four,  and  an  equal 
negative  valence,  both  of  approximately  equal  strength.  The 
first  element  on  the  back  of  the  cylinder  is  more  negative  than 
positive,  and  has  a  positive  valence  of  five,  and  a  negative  valence 
of  three.  The  negative  properties  increase  until  fluorine  is 
reached  and  then  there  is  a  sharp  break  of  properties,  with  the 
change  from  the  strongly  negative,  univalent  element  fluorine, 
through  the  zero  valent  transition  element  neon,  to  the  strongly 
positive  sodium.  Thus  in  order  around  the  outer  loop  the  second 
series  of  elements  are  as  follows:  — 

Group  number 0          1          2          3          4          5          6        7 

Maximum  valence. ..  0          1          2          3          4          5          67 

Element He ,      Li       Be         B         C         N        OF 

Atomic  number 23456789 

"  After  these  comes  neon,  which  is  like  helium,  sodium  which  is 
like  lithium,  etc.,  to  chlorine,  the  eighth  element  of  the  second 
period.  For  the  third  period  the  journey  is  continued,  still  on 
the  outer  loop,  with  argon,  potassium,  calcium,  scandium,  and 
then  begins  with  titanium,  to  turn  for  the  first  time  into  the  inner 
loop.  Vanadium,  chromium,  and  manganese,  which  comes  next, 
are  on  the  inner  loop,  and  thus  belong,  not  to  main  but  to  sub- 
groups. This  is  the  first  appearance  in  the  system  of  sub-group 
elements.  Just  beyond  manganese  a  catastrophe  of  some  sort 
seems  to  take  place,  for  here  three  elements  of  one  kind,  and  there- 
fore belonging  to  one  group,  are  deposited.  The  eighth  group  in 
this  table  takes  the  place  on  the  inner  loop  which  the  rare  gases  of 
the  atmosphere  fill  on  the  outer  loop.  The  eighth  group  is  thus  a 
sub-group  of  the  zero  group. 

"  After  the  eighth  group  elements,  which  have  appeared  for  the 
first  time,  come  copper,  zinc,  and  gallium;  and  with  germanium, 
a  fourth  group  element,  the  helix  returns  to  the  outer  loop.  It 
then  passes  through  arsenic,  selenium,  and  bromine,  thus  completing 
the  first  long  period  of  18  elements.  Following  this  there  comes  a 
second  long  period,  exactly  similar,  and  also  containing  18  ele- 
ments. 

"  The  relations  which  exist  may  be  shown  by  the  folio  wing 
natural  classification  of  the  elements.  They  may  be  divided  into 
cycles  and  periods  as  follows: 


598  THEORETICAL  CHEMISTRY 

TABLE  I. 

Cycle  1  =  42  elements. 

1st  short  period He  —  F    =    8  =  2X22  elements. 

2nd  short  period Ne  —  Cl  =    8  =  2  X  22  elements. 

Cycle  2  =  62  elements. 

1st  long  period A  —  Br  =  18  =  2  X  32  elements. 

2nd  long  period Kr  —  I  =  18  =  2  X  32  elements. 

Cycle  3  =  82  elements. 

1st  very  long  period Xe  —  Eka-I  =  32  =  2  X  42  elements. 

2nd  very  long  period Nt  —  U. 

"  The  last  very  long  period,  and  therefore  the  last  cycle,  is  in- 
complete. It  will  be  seen,  however,  that  these  remarkable  rela- 
tions are  perfect  in  their  regularity.  These  are  the  relations, 
too,  which  exist  in  the  completed  system,*  and  are  not  like  many 
false  numerical  systems  which  have  been  proposed  in  the  past 
where  the  supposed  relations  were  due  to  the  counting  of  blanks 
which  do  not  correspond  to  atomic  numbers.  This  peculiar  rela- 
tionship is  undoubtedly  connected  with  the  variations  in  structure 
of  these  complex  elements,  but  their  meaning  will  not  be  apparent 
until  we  know  more  in  regard  to  atomic  structure. 

"  The  first  cycle  of  two  short  periods  is  made  up  wholly  of  outer 
loop  or  main  group  elements.  Each  of  the  long  periods  of  the 
second  cycle  is  made  up  of  main  and  of  sub-group  elements,  and 
each  period  contains  one-eighth  group.  The  only  complete  very 
long  period  is  made  up  of  main  and  of  sub-group  elements,  con- 
tains one-eighth  group,  and  would  be  of  the  same  length  (18  ele- 
ments) as  the  long  periods  if  it  were  not  lengthened  to  32  elements 
by  the  inclusion  of  the  rare  earths. 

"  The  first  long  period  is  introduced  into  the  system  by  the  in- 
sertion of  iron,  cobalt,  and  nickel,  in  its  center,  and  these  are  three 
elements  whose  atomic  numbers  increase  by  steps  of  one  while 
their  valence  remains  constant.  The  first  very  long  period  is 
formed  in  a  similar  way  by  the  insertion  of  the  rare  earths,  another 
set  of  elements  whose  atomic  numbers  increase  by  one  while  the 
valence  remains  constant. 

"  In  this  periodic  table  the  maximum  valence  for  a  group  of 

*  If  elements  of  atomic  weights  two  and  three  are  ever  discovered  then  the 
zero  cycle  would  contain  22  elements,  and  the  first  period  should  then  be 
said  to  begin  with  lithium.  Such  extrapolation,  however,  is  an  uncertain 
basis  for  the  prediction  of  such  elements. 


ATOMIC  STRUCTURE  599 

elements  may  be  found  by  beginning  with  zero  for  the  zero  group 
and  counting  toward  the  front  for  positive  valence,  and  toward 
the  back  for  negative  valence. 

"  The  negative  valence  runs  along  the  spirals  toward  the  back 
as  follows :  — 

0  -1  -2  -3  -4 

Ne  F  O  N  C 

A  Cl  S  P  Si 

"Beginning  with  helium  the  relations  of  the  maximum  theoreti- 
cal valences  run  as  follows :  — 

Case  1 .    He-F 0, 1,  2,  3,  4,  5,  6,  7,  but  does  not  rise  to  8.   Drops  by  7  to  0. 

Ne-Cl ...   0, 1,  2,  3,  4,  5,  6,  7,  but  does  not  rise  to  8.    Drops  by  7  to  0. 
Case  2.    A-Mn ...   0, 1,  2,  3,  4,  5,  6,  7,  8,  8.    Drops  by  7  to  1. 

Fe,  Co,  Ni. 
Case  1.    Cu-Br. 
Case  2.    Kr-Ru,  Rh,  Pd. 

"  In  the  third  increase,  the  group  number  and  maximum  valence 
of  the  group  rise  to  8,  three  elements  are  formed,  and  the  drop  is 
again  by  7  to  1. 

"  Thus  in  every  case  when  the  valence  drops  back  the  drop  in 
maximum  group  valence  is  7,  either  from  7  to  0,  or  from  8  to  1. 
This  is  another  illustration  of  the  fact  that  the  eighth  group  is  a 
sub-group  of  the  zero  group.  The  valence  of  the  zero  group  is 
zero.  According  to  Abegg  the  contra-valence,  seemingly  not 
active  in  this  case,  is  eight. 

"  In  Fig.  13  the  table  is  divided  into  five  divisions,  by 
four  straight  lines  across  the  base.  These  divisions  contain  the 
following  groups :  — 

Division 01234 

Groups 0,8         1,7         2,6         3,5         4,4 

"  The  two  groups  of  any  division  are  said  to  be  complementary. 
It  will  be  seen  that  the  sum  of  the  group  numbers  in  any  division 
is  equal  to  8,  as  is  also  the  sum  of  the  maximum  valences.  The 
algebraic  sum  of  the  characteristic  valences  of  two  complementary 
groups  is  always  zero.  In  any  division  in  which  the  group  num- 
bers are  very  different,  the  chemical  properties  of  the  elements  of  the 
complementary  main  groups  are  very  different,  but  when  the  group 
numbers  become  the  same,  the  chemical  properties  become  very  much 


600  THEORETICAL   CHEMISTRY 

alike.  Thus  the  greatest  difference  'in  group  numbers  occurs  in 
division  8,  where  the  difference  is  8,  and  in  the  two  groups  there  is 
an  extreme  difference  in  chemical  properties,  as  there  is  also  in 
division  1  between  JGroups  1  and  7. 

"  Whenever  the  two  main  groups  of  a  division  are  very  different  in 
properties,  each  of  the  sub-groups  is  quite  different  from  its  related 
main  group.  Thus  copper  in  Group  IB  is  not  very  closely  related 
to  potassium  Group  IA  in  its  properties,  and  manganese  is  not 
very  similar  to  chlorine,  but  as  the  group  numbers  approach  each 
other  the  main  and  sub-groups  become  much  alike.  Thus  scandium 
is  quite  similar  to  gallium  in  its  properties,  and  titanium  and  ger- 
manium are  very  closely  allied  to  silicon. 

"  One  important  relation  is  that  on  the  outer  cylinder  the  main 
groups  I  A,  1 1  A,  I II  A,  become  less  positive  as  the  group  number 
increases,  while  on  the  inner  loop  the  positive  character  increases  fr6m 
Group  IB  to  II B,  and  at  the  bottom  of  the  table  the  increase  from 
IIB  to  IIIB  is  considerable.  Thus  thallium  is  much  more  posi- 
tive than  mercury.  It  has  already  been  noted  that  in  the  case  of 
the  rare  earths  also  the  usual  rule  is  inverted,  that  is  the  basic 
properties  decrease  as  the  atomic  weight  increases." 

The  Octet  Theory  While  the  theory  of  atomic  structure  out- 
lined at  the  beginning  of  the  present  chapter  has  furnished  a  satis- 
factory interpretation  of  a  number  of  purely  physical  phenomena, 
its  applicability  to  chemical  phenomena  is  exceedingly  limited. 
A  theory  of  atomic  structure  originally  advanced  by  G.  N.  Lewis,* 
and  subsequently  elaborated  by  Langmuir,  f  has  been  found  ade- 
quate to  interpret  practically  all  of  the  chemical,  as  well  as  many 
of  the  physical  properties,  of  the  elements.  In  this  theory,  com- 
monly known  as  the  octet  theory  of  atomic  structure,  it  is  assume^ 
that  the  range  o  vibration  of  the  electrons  is  so  restricted  that 
they  may  practically  be  considered  as  occupying  stationary  posi- 
tions, symmetrically  distributed  about  the  atomic  nucleus.  It  is 
also  assumed,  that  the  electrons  are  arranged  in  a  series  of  concen- 
tric shells,  the  first  of  which  contains  two  electrons,  while  the  suc- 
ceeding shells  may  contain  as  many  as  eight  electrons.  These 
eight  electrons  are  assumed  to  distribute  themselves  singly  at  the 
corners  of  a  cube,  or  in  pairs  at  the  corners  of  a  regular  tetrahedron. 
In  either  case,  the  arrangement  is  assumed  to  be  symmetrical. 

*  Jour.  Am.  Chem.  Soc.,  38,  762  (1916). 
t  Ibid.  41,  868  (1919) 


ATOMIC  STRUCTURE  601 

The  fundamental  postulates  upon  which  the  octet  theory  is  based 
have  been  briefly  formulated  by  Langmuir,  as  follows: 

Postulate  I.  The  electrons  in  the  atoms  of  the  inert  gases  are 
arranged  about  the  nucleus  in  pairs,  symmetrically  placed  with 
respect  to  a  plane  passing  through  the  nucleus,  which  we  may  call 
the  equatorial  plane.  No  electrons  lie  in  the  equatorial  plane. 
There  is  an  axis  of  symmetry,  perpendicular  to  the  plane,  through 
which  four  secondary  planes  of  symmetry  pass,  forming  angles 
of  45°  with  each  other.  The  symmetry  thus  conforms  to  that  of  a 
tetragonal  crystal. 

Postulate  II.  The  electrons  in  the  atoms,  are  distributed 
through  a  series  of  concentric  spherical  shells.  All  the  shells  in  a 
given  atom  are  of  equal  thickness.  If  the  mean  of  the  inner  and 
outer  radii  be  considered  to  be  the  effective  radius  of  the  shell, 
then  the  radii  of  the  different  shells  stand  in  the  ratio,  1:2:  3  : 
4,  and  the  effective  surfaces  of  the  shells  are  in  the  ratio,  1  :  22  : 
32  :  42. 

Postulate  III.  Each  spherical  shell  is  divided  into  a  num- 
ber of  cellular  spaces.  The  thickness  of  these  cells,  measured  in 
a  radial  direction,  is  equal  to  the  thickness  of  the  shell  and  is, 
therefore,  the  same  for  all  the  cells  in  the  atom.  The  first  shell 
contains  two  cells,  obtained  by  dividing  the  shell  into  two  parts 
by  the  equatorial  plane.  The  second  shell,  having  four  times  the 
surface,  contains  eight  cells.  The  third  shell  thus  contains  eight- 
een, while  the  fourth  contains  thirty-two  cells. 

Postulate  IV.  Each  of  the  two  innermost  cells  can  contain 
only  one  electron,  but  each  of  the  other  cells  is  capable  of  holding 
two.  There  can  be  no  electrons  in  the  outside  shell  until  all  of 
the  inner  shells  contain  their  maximum  numbers  of  electrons. 
In  the  outside  shell,  two  electrons  can  occupy  a  single  cell  only 
when  all  other  cells  contain  at  least  one  electron. 

Postulate  V.  It  is  assumed  that  electrons  contained  in  the 
same  cell  are  nearly  without  effect  on  each  other.  But  the  elec- 
trons in  the  outside  layer  tend  to  align  themselves  in  a  radial 
direction  with  those  of  the  underlying  shell,  because  of  a  mag- 
netic field,  probably  always  to  be  associated  with  electrons  bound 
in  atoms. 

Postulate  VI.  When  the  number  of  electrons  in  the  outside 
layer  is  small,  the  magnetic  attraction  exerted  by  the  electrons 
of  the  inner  sheik  tends  to  predominate  over  the  electrostatic 


602  THEORETICAL  CHEMISTRY 

repulsion;  but  when  the  atomic  number  increases,  the  number  of 
electrons  in  the  outside  layer  increases,  and  the  electrostatic  forces 
gradually  become  the  controlling  factor. 

Postulate  VII.  The  properties  of  the  atom  are  determined 
not  only  by  the  number  and  arrangement  of  electrons  in  the  out- 
side layer,  but  also  by  the  ease  with  which  they  are  able  to  revert 
to  more  stable  forms,  by  giving  up  or  taking  up  electrons,  or  by 
sharing  their  outside  electrons  with  atoms  with  which  they  com- 
bine. The  tendencies  to  revert  to  the  forms  represented  by  the 
atoms  of  the  inert  gases  are  the  strongest,  but  there  are  a  few 
other  forms  of  high  symmetry,  such  as  those  corresponding  to 
certain  possible  forms  of  nickel,  palladium,  erbium  and  platinum 
atoms,  towards  which  atoms  have  a  weaker  tendency  to  revert 
(by  giving  up  electrons  only). 

Postulate  VIII.  The  very  stable  arrangements  of  electrons, 
corresponding  to  those  of  the  inert  gases,  are  characterized  by 
strong  internal,  but  unusually  weak  external,  fields  of  force.  The 
smaller  the  atomic  number  of  the  element,  the  weaker  are  these 
external  fields. 

Postulate  IX.  The  pair  of  electrons  in  the  helium  atom  rep- 
resents the  most  stable  possible  arrangement.  A  stable  pair 
of  this  kind  is  formed  only  under  the  direct  influence  of  positive 
charges.  The  positive  charges  producing  the  stable  pair  may 
be: 

(a)  The  nucleus  of  any  element. 

(b)  Two  hydrogen  nuclei. 

(c)  A  hydrogen  nucleus  together  with  the  kernel  of  an  atom. 

(d)  Two  atomic  kernels. 

These  are  listed  in  the  order  of  their  stability. 

Postulate  X.  The  next  most  stable  arrangement  of  electrons 
is  the  group  of  eight,  such  as  forms  the  outside  layer  in  atoms  of 
neon  and  argon.  We  shall  call  this  stable  group  of  eight  electrons, 
the  "  octet."  Any  atom  having  an  atomic  number  less  than 
twenty,  and  having  more  than  two  positive  charges  on  its  kernel, 
tends  to  take  up  electrons  to  form  an  octet.  The  greater  the 
charge  on  the  kernel,  the  stronger  is  this  tendency. 

Postulate  XI.  Two  octets  may  hold  one,  two,  or  sometimes 
even  three  pairs  of  electrons  in  common.  An  octet  may  share 
an  even  number  of  its  electrons  with  one,  two,  three,  or  four  other 
octets.  No  electrons  can  form  parts  of  more  than  two  octets. 


ATOMIC  STRUCTURE  603 

The  Periodic  System  and  the  Octet  Theory.  The  periodicity 
of  the  properties  of  all  of  the  elements,  including  the  rare-earths 
and  the  members  of  the  eighth  group,  can  be  satisfactorily  ac- 
counted for  by  the  octet  theory.  Lack  of  space  prohibits  our 
giving  more  than  a  brief  account  of  this  interesting  application 
of  the  octet  theory.  The  classification  of  the  elements  according 
to  the  arrangement  of  their  electrons  is  shown  in  the  table  given 
on  p.  604,  which  will  be  seen  to  bear  a  close  resemblance  to  the 
familiar  periodic  arrangement. 

The  Roman  numerals  in  the  first  column  denote  the  serial  num- 
bers of  the  outer  shell  of  the  atom  while  the  accompanying  letters 
indicate  the  outside  layer  of  this  shell.  The  numbers  in  the  sec- 
ond column  represent  the  index  number,  and  the  subscripts  to  the 
symbols  of  the  elements  denote  their  atomic  numbers  In  order 
to  find  the  number  of  electrons  in  the  unfilled  outside  layer  of  the 
atom  of  any  element,  subtract  from  its  atomic  number,  the 
preceding  index  number,  given  in  the  second  column  of  the 
table. 

A  series  of  drawings  showing  the  arrangement  of  the  electrons 
in  the  atoms  of  the  elements  of  the  first  and  second  periods  of  the 
table  is  given  in  Fig.  143. 

The  atom  of  hydrogen  has  but  one  electron  which  revolves 
about  the  positive  nucleus.  It  is  unsaturated  and,  therefore, 
tends  to  take  up  another,  electron  in  order  to  assume  the  sym- 
metrical arrangement  which  is  characteristic  of  helium.  The 
valence  of  hydrogen  is  evidently  unity. 

The  first  drawing  represents  the  atom  of  helium.  This  will  be 
seen  to  consist  of  a  nucleus  made  up  of  four  hydrogen  nuclei,  held 
together  by  two  electrons,  and  of  an  outer  shell  containing  two 
electrons.  This  probably  represents  the  most  stable  grouping 
of  electrons  possible,  and,  in  consequence,  helium  is  to  be  regarded 
as  the  most  stable  of  all  the  elements. 

The  atom  of  lithium  has  three  electrons  and  three  positive 
nuclear  charges.  Only  one  of  the  three  electrons,  however,  is 
in  the  outer  shell,  the  other  two  being  required  to  form  a  stable 
pair  within  the  shell,  similar  to  that  shown  in  the  diagram  of  the 
helium  atom.  In  all  of  the  remaining  diagrams,  only  the  elec- 
trons in  the  outer  shell  will  be  represented. 

In  beryllium  and  boron,  the  properties  are  determined  largely 
by  the  ability  of  the  atom  to  revert  to  the  form  corresponding 


604 


THEORETICAL  CHEMISTRY 


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ATOMIC  STRUCTURE 


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to  helium.     The  actual  arrangement  of  the  electrons  in  the  atoms 
of  these  elements  is  thus  of  little  significance. 

In  the  carbon  atom,  the  four  electrons  tend  to  arrange  them- 
selves at  the  corners  of  a  regular  tetrahedron,  as  shown  in  the 
diagram. 


He 


Li 


Be 


A-4 

O 


Ne 


L4-4 

Na 


£_ 
Mg 


Al 


Si 


Cl 


Fig.  143 

In  the  case  of  the  nitrogen  atom,  no  symmetrical  ar- 
rangement of  the  five  electrons  is  possible.  This  lack  of  sym- 
metry is  undoubtedly  responsible  for  the  tendency  which  nitrogen 
exhibits  to  form  a  series  of  unusual  compounds. 

While  the  properties  of  the  elements  from  lithium  to  carbon 
vary  progressively,  in  passing  from  carbon  to  nitrogen  there  is  a 
marked  discontinuity  in  properties.  The  following  may  be  cited 
as  illustrations  of  the  strong  contrast  in  the  properties  of  these  two 
elements :  The  constant  valence  of  carbon  —  the  variable  valence 
of  nitrogen;  the  high  melting-point  of  carbon  —  the  low  melting- 
point  of  nitrogen;  the  marked  inertness  and  stability  of  the 
majority  of  carbon  compounds  —  the  great  activity  and  instabil- 
ity of  almost  all  nitrogen  compounds. 

According  to  Postulate  X,  the  atom  of  an  element  having  less 
than  eight  electrons  in  its  outer  shell,  tends  to  acquire  a  sufficient 
number  of  electrons  to  complete  its  octet.  Or,  in  case  the  atom 
has  but  few  electrons  in  its  outer  shell,  it  tends  to  give  up  elec- 
trons to  the  atom  of  another  element,  thereby  completing  its 
octet.  For  example,  the  atom  of  lithium  has  only  a  single  elec- 
tron in  its  outer  shell  and,  therefore,  tends  to  complete  the  octet 
of  the  atom  of  some  other  element,  such  as  fluorine,  which  lacks 
only  one  electron  of  a  complete  octet.  The  result  of  this  redis- 
tribution of  electrons  is  the  formation  of  the  compound,  LiF. 

In  like  manner,  beryllium  has  two  electrons,  hence,  by  parting 
with  one  electron  to  each  of  two  fluorine  atoms  their  octets  will  be 
completed  and  the  compound,  BeF2,  will  be  formed. 


606  THEORETICAL  CHEMISTRY 

The  outer  shell  of  the  carbon  atom  lacks  four  electrons  of  a 
complete  octet.  In  this  case,  the  atom  may  either  give  up  its 
four  electrons,  or  acquire  four  extra  electrons  from  some  other 
atom,  thereby  completing  its  octet.  For  example,  a  carbon  atom 
can  give  up  its  four  electrons  to  each  of  two  oxygen  atoms,  thus 
completing  their  octets  and  forming  a  molecule  of  carbon  di- 
oxide, or,  it  can  take  up  an  electron  from  each  of  four  hydrogen 
atoms,  thereby  completing  its  own  octet,  and  forming  a  molecule 
of  methane,  CH4. 

Furthermore,  if  an  atom  is  deprived  of  the  opportunity  of  com- 
pleting its  octet  at  the  expense  of  the  atom  of  another  element, 
it  may  share  a  pair  of  electrons  with  another  atom  of  the  same 
element.  For  example,  two  oxygen  atoms  may  complete  their 
octets  by  sharing  a  pair  of  electrons  between  them,  thereby  form- 
ing a  single  molecule  of  oxygen,  02. 

Turning  now  to  the  second  period  of  the  table,  we  begin  with 
neon,  which  has  a  complete  octet  in  its  outer  shell,  and  hence  shows 
no  tendency  to  give  up,  or  acquire  electrons.  This  is  in  complete 
accord  with  the  well-known  inertness  of  the  element. 

The  next  succeeding  element  is  sodium,  which  should  resemble 
its  analogue,  lithium,  in  the  arrangement  of  its  eleven  electrons. 
Since  two  electrons  are  required  to  stabilize  the  nucleus,  and 
eight  more  electrons  are  required  to  complete  the  first  octet,  it 
follows,  that  the  remaining  electron  must  become  the  first  member 
of  the  next  succeeding  octet.  The  inner  stable  pair  of  electrons, 
and  the  electrons  constituting  the  first  octet,  are  without  influence 
on  the  properties  of  the  element.  This  it  will  be  observed,  is  in 
accordance  with  Postulate  VII,  which  states  that  the  properties  of 
the  atom  are  determined  by  the  number  and  arrangement  of  the 
electrons  in  the  outer  layer.  The  sodium  atom  has  no  tendency 
to  acquire  seven  other  electrons,  but  readily  gives  up  its  one  elec- 
tron to  complete  the  octet  of  the  atom  of  some  other  element. 
In  the  event  of  its  losing  one  electron,  the  atom  of  sodium  will 
be  left  with  a  positive  charge,  and  may  be  regarded  as  a  free  so- 
dium ion. 

The  arrangement  of  the  electrons  in  the  outer  shells  of  the 
remaining  elements  of  the  second  period  resembles  that  of  their 
analogues  in  the  first  period.  For  example,  the  atom  of  silicon 
like  the  atom  of  carbon,  can  either  give  up  four  electrons  to  com- 
plete the  octet  of  the  atom  of  some  other  element,  or  it  can  take 


ATOMIC  STRUCTURE  607 

Up  four  additional  electrons  and  thereby  complete  its  own  octet. 
The  last  element  of  the  period,  chlorine,  has  seventeen  electrons, 
and  by  adding  one  more  electron,  the  second  octet  will  be  com- 
pleted, thereby  again  producing  an  arrangement  of  electrons  which 
is  characteristic  of  the  inert  gases  On  referring  to  the  table, 
we  find  that  this  new  configuration  of  electrons  corresponds  to 
argon,  the  first  member  of  the  next  period.  In  this  manner  the 
theory  may  be  applied  to  the  remaining  elements  in  the 
table. 

The  Octet  Theory  and  Valence  The  octet  theory  also  fur- 
nishes an  interesting  interpretation  of  valence,  which  is  applic- 
able to  both  inorganic  and  organic  compounds.  Let  e  be  the 
total  number  of  ava  lable  electrons  in  the  shel  s  of  the  atoms  form- 
ing a  given  molecule.  Let  n  be  the  number  of  octets  formed  by 
their  combination,  and  let  p  be  the  number  of  pairs  of  electrons 
held  in  common  by  the  octets.  For  every  pair  of  electrons  held 
in  common,  there  is  a  saving  of  2  p  in  the  number  of  electrons 
needed  to  form  the  octets.  Therefore,  we  have 

e  =  8  n  -  2  p  (3) 

It  is  usually  found  more  convenient  to  use  the  equation  in  the  form, 

p  =  l/2(8n-e).  (4) 

When  a  hydrogen  nucleus  holds  a  pair  of  electrons  in  common  with 
an  octet,  this  pair  should  not  be  included  in  determining  the  value 
of  p,  since  it  does  not  result  in  any  saving  in  the  numbers  of  elec- 
trons required  to  form  the  octets.  In  order  to  determine  e,  we 
add  together  the  number  of  available  electrons  in  the  outside 
shells  of  all  the  constituent  atoms.  Thus,  for  every  hydrogen 
atom  we  add  1,  for  every  lithium  atom,  1;  for  every  beryl- 
lium atom,  2;  for  every  carbon  atom,  4;  and  for  every  oxygen 
atom,  6.  Equation  (4)  gives  us  information  as  to  the  ways  in 
which  the  octets  in  a  given  molecule  can  be  arranged.  It  applies 
to  all  compounds  whose  atoms  are  held  together,  either  because 
their  octets  share  electrons,  or  because  electrons  have  been  trans- 
ferred from  one  atom  to  another  in  order  to  complete  the 
octets. 

The  arrangement  of  the  electrons  in  the  molecules  of  several 
familiar  compounds  is  shown  in  the  following  diagrams,  Figs. 
144  to  147. 


608 


THEORETICAL  CHEMISTRY 

H, 


H2O 
n-l.e-8.p-0 

Fig.   144 

<£^> 


H3P04 
n=5.  e=32,  p=4 


£.         r 

•f        , 

( 

y 

>    ''     ' 
( 

f  —  : 

O 

N 

A 

/° 

7 

i 

N 

c 
205 

0 

o 

•  ' 

N 

\ 

O 

HNO3 
n-4,  e=24,  p=4 


Fig.  146 


Fig.  147 


Physical  Properties  and  Electron  Configuration.  Further  evi- 
dence of  the  fundamental  correctness  of  the  octet  theory  is  fur- 
nished by  the  fact,  that  a  striking  similarity  is  found  to  exist  be- 


N 


N 


/i 

/i 

/ 

o 

IN  ,__ 

j           £ 

1 
--&-  

i  o 

Fig.  148 

tween  the  physical  properties  of  compounds  whose  molecules 
resemble  each  other  in  the  number  and  configuration  of  their 
electrons.  For  example,  carbon  dioxide  and  nitrous  oxide,  two 
gases  of  dissimilar  chemical  properties,  are  found  to  resemble 
each  other  very  closely  in  their  physical  properties.  A  molecule 
of  each  gas  contains  sixteen  electrons  arranged  around  three  atomic 
nuclei,  as  shown  in  Fig.  148.  It  will  be  seen  that  the  central  atom 
in  each  molecule  shares  four  nairs  of  electrons,  two  pairs  being 


ATOMIC   STRUCTURE 


609 


shared  with  each  of  the  adjacent  octets.  The  following  table  of 
physical  constants  of  the  two  gases  shows  the  remarkable  re- 
semblance between  their  various  physical  properties. 

PHYSICAL  CONSTANTS  OF  N2O  AND  CO2 


Constant 

N2O 

C02 

Boiling-point,  Abs  

183  2° 

194  7° 

Critical  temperature,  Abs.  

309  5° 

304° 

Critical  pressure,  Atmos 

71  65 

72  85 

Critical  density 

0  454 

0  448 

Density  of  liquid,  —20°. 

0  996 

1  031 

Density  of  liquid,  +10°.    . 

0  856 

0  858 

Solubility  in  water,  5°  

0  205 

0  277 

Viscosity,  0°,  77  X  107  

1408 

1414 

Refractive  index  of  liquid.  .    . 

1  193 

1  190 

Dielectric  constant  of  liquid,  0°.  . 

1  598 

1  582 

Magnetic    susceptibility  of    gas  at  40  atmos. 
X  10-6,  16°.  .. 

0  12 

0  12 

The  data  of  the  preceding  table,  taken  together  with  all  of  the 
other  available  evidence,  make  it  appear  that  the  postulates 
upon  which  the  octet  theory  is  based  are  fundamentally  correct. 
Although  future  research  will  undoubtedly  necessitate  modifica- 
tion of  the  theory  in  some  of  its  minor  details,  it  seems  most  un- 
likely that  it  will  be  wholly  superseded  by  some  inherently  differ- 
ent conception  of  atomic  and  molecular  structure. 


INDEX  OF  AUTHORS 


Abegg,  204,  599 

Adams,  206 

Adler,  520 

Adrian!,  347 

Alexieeff,  157 

Amagat,  24,  33 

Andrews,  54,  65 

Arrhenius,  189,  211,  868,  376,  397, 

412,  432,  435,  439 
Aston,  427,  565 
Avogadro,  8,  16,  23,  27 

Babo,  187 

Baker,  382,  540 

Baly,  123 

Bancroft,  382,  494,  533 

Barlow,  90 

Barry,  291 

Barus,  255 

Bates,  394,  437 

Baume,  68 

Baxter,  584 

Beans,  452,  467 

Beattie,  487 

Beccaria,  389 

Bechhold,  250 

Beckmann,  204 

Becquerel,  E.,  537,  544 

Becquerel,  H.,  569 

Beer,  124,  534 

Bein,  488 

Benedict,  277 

Bennett,  520 

Bergmann,  302 

Berkeley,  Earl  of,  182,  186,  195,  220 

Bernoulli,  27 

Berthelot,  18,  62,  275,  291,  298,  303, 

314,  319 

Berthollet,  4,  302 
Berzelius,  8,  14,  18,  390 


Bigelow,  198,  378 

Biltz,  252,  436 

Bingham,  72,  82 

Biot,  116 

Bjerrum,  497 

Blagden,  202 

Blake,  245 

Bodenstein,  309  537 

Boguski,  375 

Bohi,  452 

Bohr,  586 

Boltwood,  577,  581 

Boltzmann,  27,  527 

Bottger,  422,  502 

Boyle,  2,  6,  16,  21,  273 

Bragg,  W.  H.,  92,  587 

Bragg,  W.  L.,  92,  587 

Brann,  485 

Brauner,  554 

Bray,  385 

Bredig,  270,  386,  412,  449,  494 

Bridgman,  337 

Bronsted,  568 

Brown,  232 

Bruhl,  109,  112 

Brimi,  242 

Buckingham,  146 

Budde,  32 

Bunsen,  31,  153,  534,  538 

Burgess,  277,  538 

Burnett,  540 

Burton,  195,  229,  245 

Cailletet,  65,  70 
Callendar,  185 
Cannizzaro,  16. 
Carlisle,  389 
Carnot,  140 
Chapman,  538 
Ciamician,  543 


611 


612 


INDEX  OF  AUTHORS 


Clapeyron,  142 

Clark,  504 

Claude,  67 

Clausius,  27,  136,  144,  397 

Clement,  46,  381 

Clowes,  254 

Coombs,  76 

Cotton,  244 

Cottrell,  200 

Creighton,  72 

Crookes,  557 

Gumming,  481 

Curie,  570,  580  583,  585 

Dale,  108 

Dalton,  1,  5,  7,  15,  18,  151,  154 

Davis,  291 

Davy,  389 

Deacon,  382 

Debierne,  31,  583 

Debray,  322 

Debye,  96 

De  Chancourtois,  547 

De  Forcrand,  328 

Denham,  449,  488 

Derr,  540 

Desch,  125,  171 

Desormes,  46,  381 

Deville,  39,  312 

Dewar,  65 

Dobereiner,  547 

Draper,  539 

Drude,  128 

Duane,  588 

Duclaux,  240 

Dulong,  10,  13,  97,  527 

Dumas,  546 

Dunphy,  196 

Dutoit,  427 

Eder,  542 
Einstein,  237 
Elliott,  251 
Ende,  485 
Eotvos,  76 

Fajans,  589 

Falk,  212,  223,  402,  408 


Fanjung,  240 

Faraday,  7,  64,  121,  127,  393 

Farmer,  450 

Faure,  506 

Fernbach,  386 

Fery,  530 

Fletcher,  238 

Forbes,  17,  475 

Fraenkel,  449 

Frazer,  185,  192,  216,  385 

Freudenberg,  521 

Freundlich,  247,  256,  263 

Friedrich,  91,  427 

Fuchs,  512 

Gay-Lussac,  8,  16,  21,  135 

Geddes,  262 

Geiger,  576,  586 

Geoffroy,  302 

Getman,  220,  424,  485,  497 

Ghosh,  497 

Gibbons,  424 

Gibbs,  138,  266,  315,  332 

Gladstone,  108 

Goodwin,  477 

Gouy,  232 

Graham,  30,  226,  268 

Gray,  18,  574 

Grotthuss,  396.  533 

Grove,  396 

Grover,  584 

Guldberg,  4,  71,  304,  323 

Guthrie,  343 

Guye,  18,  71,  121 

Haber,  380,  382 

Hall,  217,  584,  595 

Hampson,  66 

Hardy,  245,  248 

Harkins,  79,  217,  415,  568,  586,  592, 

594 

Harned,  497 

Hartley,  123,  182,  186,  195 
Hautefeuille,  309 
Haiiy,  90 
Helmholtz,  131,  138,  315,  459,  473, 

487 
Henderson.  481 


INDEX  OF  AUTHORS 


613 


Henry,  154,  469 
Hess,  277,  293 
Heuse,  195 
Hevesy,  568 
Heycock,  355 
Heydweiller,  421 
Higgins,  277 
Hildebrand,  73 
Him,  431 
Hittorf,  398,  403 
Hofmeister,  258 
Holborn,  487,  530 
Horstmann,  325 
Hudson,  452 
Hulett,  168 
Hull,  92,  525 
Hurter,  382 

Isambert,  351 

Jahn,  460,  488 
Jones,  219 
Joule,  131,  135 
Jurin,  74 

Kahlenberg,  395 

Kanolt,  452 

Kekule,  112 

Kendall,  497 

Keyes,  485 

Kirchhoff,  526 

Knipping,  91 

Knoblauch,  374 

Kniipffer,  494 

Koelichen,  434 

Kohlrausch,  405,  409,  412,  418,  421, 

451,  487,  501 
Konowalow,  160 
Kopp,  13,  39,  102 
Kothner,  16 
Kraus,  437,  485 
Kroenig,  27 
Kundt,  46 
Kurlbaum,  530 

Laborde,  585 
Lacey,  485 
Lamb,  385,  484 


Landolt,  3,  115 

Langbein,  291 

Langmuir,  600 

Laplace,  47,  277 

Larmor,  372 

Larson,  484 

Laue,  91 

Lavoisier,  2,  273,  277 

Le  Bas,  102,  129 

Lebedew,  525 

Le  Bel,  118,  121 

Le  Blanc,  512,  516,  520 

Le  Chatelier,  45,  287,  298,  529 

Lehmann,  99 

Lemoine,  309 

Lenard,  531,  559 

Lewis,  G.  N.,  97,  146,  288,  485,  600 

Lewis,  W.  C.  McC.,  146 

Ley,  449 

Liebig,  380 

Lille,  241 

Linde,  66 

Lindemann,  98,  568 

Linder,  230,  238,  243,  246 

Linhart,  497 

Lippmann,  469 

Lodge,  417 

Lohnstein,  79 

Loomis,  204 

Lorentz,  109 

Lorenz,  108,  424,  452 

Lotz,  186 

Lovelace,  192,  216 

Lubs,  504 

Liideking,  259 

Lummer,  527 

Lunden,  449,  452 

Luther,  540 

Macdougall,  146 

Maclnnes,  481,  484,  487,  495,  520 

Marignac,  5,  15 

Mathews,  198 

Mathias,  70 

Maxwell,  27 

Mayer,  43,  131,  277 

Mendeleeff,  3,  15,  548,  552 

Menschutkin,  377 


614 


INDEX  OF  AUTHORS 


Meyer,  475 
Meyer,  L.,  548,  551 
Meyer,  V.,  34,  37,  50 
Millard,  403 
Millikan,  24,  238,  562 
Milner,  497 
Mitscherlich,  13,  90 
Morgan,  79 
Morley,  15,  35 
Morse,  181,  184 
Mortimer,  487 
Moseley,  588 
Mouton,  244 
Myrick,  185 

Natanson,  E.  &  L.,  42 

Natterer,  32 

Nernst,  38,  72,  98,  131,  208,  287,  299, 

317,  323,  329,  333,  426,  462,  473, 

478,  484,  536 
Neumann,  13 
Neville,  355 
Newberry,  519 
Newlands,  548 
Newton,  6,  302 
Nichols,  525 
Nicholson,  3^9 
Nicholson,  J.  W.,  586 
Nollet,  175 
Nordlund,  237 
Noyes,  212,  217,  223,  371,  375,  402, 

408,  420,  452,  485,  490,  495 

Oakes,  452,  467 

Ogg,  486 

Ohm,  392 

Olszewski,  65 

Onnes,  68 

Ostwald,  W.,  77,  101,  152,  178,  218, 

266,  273*,  368,  377,  398,  430,  442, 

458,  544 
Ostwald,  Wo.,  232 

Palitzsch,  504 
Palmaer,  463 
Pappada,  242 
Partington,  146 
Pasteur,  116 


Patrick,  127 

Pauli,  246 

Pearce,  487 

Pebal,  39 

Perkin,  122 

Perkins,  146 

Perrin,  233,  236,  243,  558 

Peters,  494 

Petit,  10,  13,  97,  527 

Pfeffer,  175 

Philip,  153,  219 

Pickering,  254 

Pictet,  65 

Picton,  230,  238,  243,  246 

Planck,  481 

Plante,  506 

Poiseuille,  81 

Pope,  90 

Porter,  220 

Prideaux,  454 

Pringsheim,  527 

Proust,  4 

Prout,  15,  546,  587 

Pulfrich,  106 

Quincke,  239,  243 

Ramsay,  50,  71,  77,  80, 151,  574 

Randall,  288,  414 

Raoult,  187,  190,  198,  204,  208 

Rayleigh,  192,  529 

Read,  201 

Redgrove.  292 

Regnault,  34 

Reicher,  368 

Reinders,  347 

Reinitzer,  99 

Reuss,  243 

Richards,  15,  19,  60,  76,  91,  277,  291, 

394,  475,  584 
Richter,  4 
Rigollot,  545 
Roberts-Austen,  171 
Rodewald,  259 
Roentgen,  569 
Roozeboom,  344 
Rosanoff,  196 
Roscoe,  163,  534,  538 


INDEX  OF  AUTHORS 


615 


Rose,  303 

Rosenstein,  453 

Royds,  574 

Rudolphi,  437 

Rupert,  485 

Rutherford,  571,  574,  577,  580,  586 

Sabatier,  383,  384 

Sachanov,  426 

Sackur,  146 

Salm,  454 

Sargent,  481,  483 

Scatchard,  219 

Schlesinger,  425 

Schmidt,  158 

Scholl,  545 

Schreinemakers,  351 

Schroader,  257 

Sease,  216 

Senderens,  383 

Senier,  2 

Seubert,  554 

Sheppard,  250,  538,  544 

Shields,  77,  451 

Siedentopf ,  229 

Smiles,  129 

Snell,  105 

Soddy,  3,  567,  578,  582,  588,  591 

Sorensen,  501,  504 

Stas,  15,  546 

Steele,  418 

Stefan,  527 

Steno,  87 

Stewart,  217,  591 

St.  Gilles,  Pean  de,  303,  314 

Stokes,  235,  531,  561 

Storch,  485 

Strutt,  581 

Stull,  395 

Sutherland,  497 

Svedberg,  271 

Sweet,  250 

Swietoslawski,  110,  292 

Tammann,  337 
Than,  41 
Thiele,  111 
Thilorier,  64 


Thompson,  520 

Thomsen,  291 

Thomson,  J.  J.,  235,  391,  426,  458, 

558,  565,  571,  586 
Thornton,  292 
Tiede,  16 
Traube,  104,  175 
Trouton,  72 
Tyndall,  229 

Urbain,  531 

van  Bemmelen,  256 

van  der  Stadt,  372 

van  der  Walls,  32,  57,  61,  73,  102, 

155 
van't  Hoff,  118,  151,  170,  179,  191, 

198,  208,  211    308,  316,  376,  412, 

437,  491 
Vinal,  394 
Vogel,  540 
Volta,  389,  457 

Waage,  4,  304,  323 

Walden,  422 

Walker,  3,  69,  191,  374 

Walpole,  484,  504 

Warder,  368 

Wartenberg,  38 

Washburn,  195,  201,  403,  420,  488 

Weber,  12 

Weimarn,  232,  267 

Weiss,  86 

Wells,  17 

Wenzel,  302 

Werner,  221 

Whetham,  418 

Whitney,  245,  375 

Wiedemann,  243,  259,  525 

Wien,  527 

Wiener,  232 

Wildermann,  545 

Williamson,  397 

Wilson,  C.  T.  R.,  559,  572 

Wilson,  E.  D.,  592 

Winkler,  552 

Winther,  542 

Wood,  531 


616  INDEX  OF  AUTHORS 

Wormann,  296  Yeh,  484 

Wright,  349  Young,  61,  71,  170 

Wroblewski,  65 

Wullner,  187  Zsigmondy,  228,  250 


INDEX  OF  SUBJECTS 


Abnormal  solutes,  210,  412 
Abnormal  vapor  densities,  38,  210 
Absorption  coefficient  of  Bunsen,  153 
Absorption  of  electric  vibrations,  127 
Absorption  and  emission,  526 
Absorption  index,  533 

for  electric  waves,  128 
Absorption  of  light,  533 
Absorption  ratio,  534 
Absorption  spectra,  123,  218 
Accumulators,  505 
Actinometers,  539 
Activity  and  ionization,  494 
Adiabatic  processes,  139 
Adsorption,  260,  381 

of  gases,  261 

in  solutions,  262 

isotherms,  261 
Alloys,  352 
Amicrons,  228 
Anomalous  absorption,  128 
Assimilation  of  carbon  dioxide,  542 
Association,  210 

factor  of,  77 

in  solution,  210 
Atomic  heats,  table  of,  11 

heats,  variation  of,  12 

number,  555,  587 

structure,  586,  592 

theory,  6 

volumes,  102 

weights,  7,  10,  14,  16,  18,  63,  567, 

594 

Autocatalysis,  379 

Avogadro's  constant,    23,    233,    564, 
577 

hypothesis,  8,  16,  30,  38 


Basicity  of  organic  acids,  441 
Bimolecular  reactions,  367 


Binary  mixtures,  vapor  pressure  of, 

158 

Black  body,  526 
Boiling-point  apparatus,  198 
Boiling-point,  binary  mixtures  having 

maximum,  162 
binary  mixtures  having  minimum, 

162 

constant,  196 
elevation  of,  195 
elevation  of,  and  osmotic  pressure, 

197 

of  a  liquid,  69,  71,  102 
Boiling-points,  and  mixtures  of  im- 
miscible   liquids    of    solutions, 
experimental    determination   of, 
198 
Brownian  movement,  232 

recent  investigations  of,  237 
Buffer  solutions,  503 

Calorimeter,  combustion,  275 
Capillary,  electrometer,  468 
Carrier,  381 
Catalysis,  377 

criteria  of,  378 

ferments  and  enzymes  in,  386 

mechanism  of,  380 

negative  and  positive,  378 

some  applications  of,  382 
Catalytic  action  of  water,  382 
Cataphoresis,  243 

Cathode     particle,     charge     carried 
by,  561 

velocity  of,  560 
Cathode  rays,  557 

properties  of,  557 
Cathode-luminescence,  532 
Cells,  chemical,  490 

galvanic,  457 


617 


618 


INDEX  OF  SUBJECTS 


Cells,  gas,  498 

photoelectric,  544 

reversible,  458 

standard,  467 

storage,  505 
Change  of  state,  143 
Chemical  cells,  490 

equivalent,  6 

kinetics,  360 

Chemi-luminescence,  532 
Colloidal  solutions,  14^ 

density  of,  238 

electrical  conductance  of,  245 

osmotic  pressure  of,  239 

preparation  of,  268 

surface  tension  of,  239 

viscosity  of,  239 
Colloids,  226 

molecular  weight  of,  242 

nomenclature,  227 

protective,  250 

surface  energy  of,  264 
Components,  332 
Compounds,  2 
Compressibilities  of  gases,  18,  32,  55 

of  solids,  91 

Concentration  elements,  475 
Conductance,  electrical,  389 

determination  of,  405 

of  different  substances,  408 

of  difficultly-soluble  salts,  421 

equivalent,  404 

of  fused  salts,  423 

at    high    pressures    and   tempera- 
tures, 420 

and  ionization,  412 

molar,  404 

of  non-aqueous  solutions,  424 

of  pure  liquids,  422 

specific,  404 

temperature  coefficient  of,  418 
Corresponding  states,  60 
Coulometer,  395 
Co-volume,  104 

Critical  constants,  method  of,  63 
Critical  pressure,  54 

solution  temperature,  157 

temperature,  54,  71,  78 


Critical  pressure,  volume,  54 
Crdoke's  dark  space,  556 
Cryohydrates,  341 

Crystal  form  and  chemical  composi- 
tion, 89 

Crystalline  precipitates,  267 
Crystallization,  13 

internal,  347 

methods  for  colloids,  269 

water  of,  221 
Crystalloids,  226 
Crystals,  liquid,  99 

mixed,  13 

properties  of,  88 

structure,  91 
Cycle,  Carnot's,  140 
Cycles,  135 

Decomposition  potentials,  513 
Degree  of  freedom,  333 
Dehydrogenation  process  of  Sabatier, 

384 

Density  of  colloidal  solutions,  238 
Densities,  limiting,  18,  63 
Dialysis,  226 
Dielectric  constants,  127 
Dilution,  heat  of,  283 

law  (Ostwald),  430 
Disintegration  series  of  uranium,  580 
Disintegration  theory,  577 
Disperse  systems,  227 
Dispersion  coefficient,  267 
Dispersion,  degree  of,  227 
Dispersoids,  228 

classification  of,  231 
Displacement  law  of  Wien,  527 
Dissociation,  39,  210 

constant,  311 

degree  of,  41,  214 

hydrolytic,  444 

and  lowering  of  vapor  pressure,  215 

of  solids,  heat  of,  328 

in  solution,  211 

of  water,  421 

Distillation  of  binary  mixtures,  161 
Distillation,  fractional,  161 

steam,  167 
Distribution,  coefficient  of,  329 


INDEX  OF  SUBJECTS 


619 


Distribution  of  a  solute  between  two 
immiscible  solvents,  329 

Efficiency  of  heat  engines,  141 

Effusion,  30 

Elasticity,  95 

Electrical  conductance,  389 

of  colloidal  solutions,  245 

determination  of,  405 
Electrical  double  layer,  463 
Electrical     methods     for     colloids, 

270 

Electrical  theory  of  matter,  556 
Electrical  units,  392 
Electrochemical  equivalent,  394 
Electrode,  dropping,  472 

hydrogen,  473,  501 

normal,  471 

null,  472 

unpolarizable,  513 
Electrodes,  of  first  type,  477 

of  second  type,  477 
Electroendosmosis,  242 
Electro-luminescence,  532 
Electrolysis,  393 

and  polarization,  511 

primary    decomposition    of   water 

in,  520 

Electrolyte,  215 
Electrolytes,  classification  of,  216 

ionization  of  strong,  436 

mixtures  of,  435 

solutions  of,  210 
Electrolytic  dissociation,   theory  of, 

212 

Electrolytic  equilibrium  and  hydroly- 
sis, 430 

Electrolytic  separation  of  metals,  521 
Electrometer,  capillary,  468 
Electrometric  titration,  502 
Electromotive  force,  457 

of  concentration  cells  with  trans- 
ference, 479 

of     concentration     cells     without 
transference,  481 

measurement  of,  465 
Electron  configuration  and  physical 
properties,  608 


Electron  theory,  556 
Elements,  2 

classification  of,  546,  604 

concentration,  475 

oxidation  and  reduction,  492 

periodic  series  of,  550 

proto-,  554 

Elevation  of  boiling  point,  195 
Emanation  of  radium,  574 
Emission  and  absorption,  526 
Emulsifiers,  254 
Emulsions,  228,  253 
Emulsoids,  228 

action  of  heat  on,  249 
Energy,  130 

radiant,  524 

relation    between    chemical    and 
electrical,  459 

source  of  radiant,  525 

distribution  of,  in  spectrum,  529 
Entropy,  145 
Equation  of,  Bates,  438 

Berthelot,  62 

Clapeyron,  142 

Clausius,  144 

Einstein,  237 

Gibbs,  266 

Gibbs-Helmholtz,  138,  315,  459 

Kraus,  438 

Nernst-Lindemann,  98 

Rudolphi,  438 

van  der  Waals,  32,  56,  73 

van't  Hoff,  437 

Equations,  thermo chemical,  274 
Equilibrium  constant,  305,  430 
Equilibrium,  heterogeneous,  322 

homogeneous,  302 

in  homogeneous  gaseous  systems, 
309 

in  liquid  systems,  313 
Equivalent  conductance,  404 
Equivalent,  electrochemical,  394 
Etch  figures,  89 
Eutectic  mixture,  point,  355 
Expansion,  isothermal,  133 

latent  heat  of,  142 
External  phase,  253 
Extinction  coefficient,  534 


620 


INDEX  OF  SUBJECTS 


Faraday  dark  space,  557 

Ferments  and  enzymes  in  catalysis, 

386 

Ferments,  inorganic,  385 
Flowing  junctions,  484 
Fluidity,  72,  82 
Fluorescence,  531 
Formation,  heat  of,  280 
Fractional  distillation,  162 
Freezing-point  apparatus,  204 
Freezing-point  constant,  202 
Freezing-point    depression,    and    os- 
motic pressure,  203 
Freezing-point,  lowering  of,  201,  212 
experimental  determination  of,  204 
Fusion,  heat  of,  85 

Galvanic  cells,  457 

Gas  cells,  498 

Gas,  ideal  or  perfect,  24 

Gaseous     systems,     equilibrium     in 

homogeneous,  309 
Gases,  21 

adsorption  of,  261 

effusion  of,  30 

ionization  of,  571 

liquefaction  of,  64 

specific  heat  of,  49 
Gelation,  228 
Gels,  227 

characteristics  of,  254 

elastic,  257 

elasticity  of,  255 

hydration  and  dehydration  of,  256 

non-elastic,  256 

physical  properties  of,  255 
Gold-number,  251 
Goniometer,  87 

Heat,  atomic,  11 

specific,  10,  42,  45 
Heat  capacity  of  solids,  95 
Heat  theorem  of  Nernst,  318 
Heat  of,  combustion,  291 

dilution,  283 

dissociation  of  solids,  328 

formation,  280 

fusion,  85 


Heat  of,  hydration,  283 

imbibition,  259 

ionization,  297,  439,  497 

neutralization,  294 

reaction,  variation  with  tempera- 
ture, 289 

solution,  282 

vaporization,  72 
Helix  of  de  Chancourtois,  547 
Heterogeneous  equilibrium,  322 
Heterogeneous  reactions,  velocity  of, 

375 

Heterogeneous  systems,  322 
Homogeneous  equilibrium,  302 
Hydrates,  326 
Hydration,  219,  403 

and  dehydration  of  gels,  256 

heat  of,  283 

Hydrogen  electrode,  501 
Hydrogen-helium  system  of  atomic 

structure,  592 
Hydrogenation  process  of  Sabatier, 

383 
Hydrolysis,  444 

and  electrolytic  equilibrium,  430 

electrometric  determination  of,  488 

experimental  determination  of,  448 

Imbibition,  heat  of,  259 

in  solutions,  260 

velocity  of,  258 
Indicators,  theory  of,  452 
Induction,  period  of,  538 
Inorganic  ferments,  386 
Internal  phase,  254 
Internal  pressure  of  liquids,  73 
Ionic    hydration    and    transference 

numbers,  403 
Ionic  product,  440 
Ionization  and  activity,  494 
Ionization  and  conductance,  412 
Ionization  constant,  430 

of  water,  451 
Ionization  of  gases,  571 
Ionization,  degree  of,  413 

heat  of,  297,  439,  497 

influence  of  substitution  on,  442 

of  salts,  acids,  and  bases,  414 


INDEX  OF  SUBJECTS 


621 


lonization  of,  strong  electrolytes,  436 

of  water,  500 

Ionizing  power  of  solvents,  426 
lonogens,  215 
Ions,  213,  393,  571 

absolute  velocity  of,  416 

complex,  217 

existence  of  free,  396 

intermediate,  217 

migration  of,  397 
Isentropics,  145 
Isobares,  591 

Isochore,  reaction,  315,  439 
Isohydric  solutions,  435 
Isomerism,  14 
Isomorphism,  13 
Isotherm,  reaction,  306 

derivation  of,  307 
Isothermal  expansion,  133 
Isothermal  processes,  139 
Isothermals,  54 

of  carbon  dioxide,  54 
Isotopes,  567,  591 

Kinetic  equation,  deductions  from,  2j) 

derivation  of,  27 
Kinetic  theory  of  gases,  27,  49 
Kinetics,  chemical,  360 

Labile  solutions,  169 

Latent  heat  of  expansion,  142 

Law  of,  Beer,  124,  534 

Boyle,  16,  21,  27,  29 

Cailletet  and  Mathias,  70 

combining  proportions,  5 

conservation  of  mass,  3 

constant  heat  summation,  277 

Dalton,  151,  329 

definite  proportions,  4 

dilution,  430,  437 

Dulong  and  Petit,  11,  552 

Faraday,  393 

gases,  deviations  from,  24 

Gay-Lussac,  8,  21,  27 

Graham,  30 

Grotthuss,  533 

Guldberg  and  Waage,  304,  322 

Henry,  154,  329 


Law  of,  Hess,  277,  294 

Jurin,  74 

Kirchhoff,  526 

Kohlrausch,  409 

Lavoisier  and  Laplace,  277 

mass  action,  304,  322 

Mitscherlich,  13 

molecular  displacement,  236 

multiple  proportions,  5 

Neumann,  13 

octaves,  548 

Poiseuille,  81 

Ramsay  and  Young,  71 

Raoult,  187 

Stefan  and  Boltzmann,  527 

Stokes,   (falling  drops),  235,  413, 
561 

Stokes,  (fluorescence),  531 

Trouton,  72 

volumes,  8,  21,  27 

Wien,  527 

Light,  action  on  silver  halides,  540 
Limiting  densities,  18 
Liquefaction  of  gases,  64 
Liquid  air  machines,  66 
Liquid  systems,  equilibrium  in,  313 
Liquid  crystals,  99 
Liquids,  characteristics  of,  53 

refractive  power  of,  105 

vapor  pressure  of,  68 
Luminescence,  525,  530 

Magnetic  rotation,  121,  219 
Mass  action,  4,  302 

law  of,  304,  322 
Mass,  conservation  of,  3 
Mass  spectrograph,  565 
Mass  spectrum,  566 
Maximum  work,  principle  of,  298 
Mean  free  path,  27,  32,  60 
Mechanical  equivalent  of   heat,  43, 

132 

Membranes,  semi-permeable,  174 
Metastable  solutions,  169 
Microns,  227 
Migration  of  ions,  397 
Migration  of  suspensoids,  244 

velocity  of,  244 


622 


INDEX  OF  SUBJECTS 


Molar  conductance,  404 
Molar  volume,  9 
Molecular  attraction,  59,  73 

constitution,  101 

diameters,  60 

displacement,  law  of,  236 

gas  constant,  22 

heat,  44 

magnetic  rotation,  122 

refraction,  108 

rotation,  115 

vibrations,  96,  125 

volume,  101 

weight,  9,  33,  7B,  78 

weight  by  boiling-point  method, 
196 

weight  by  freezing-point  method, 
202 

weight  of  colloids,  242 

weight  in  solution,  207 
Mol-number,  214 
Mutual  precipitation,  252 

Negative  glow,  556 
Neutralization,  heat  of,  294 
Non-dissociating  solvent,  solution  of 

solid  in,  330 

Normal  electrode  potential,  484 
Normal  electrodes,  471 
Normal  potential  of  a  process,  494 

Occlusion,  171 
Octaves,  law  of,  548 
Octet  theory,  600 

and  valence,  607 
Optical  activity,  116 

anomaly,    exaltation,    depression, 
111 

sensitizatlon,  541 
Orthobaric  volume,  62 
Osmotic  pressure,  174,  211 

apparatus,  181 

and  boiling-point  elevation,  197 

of  colloidal  solutions,  239 

and  electrolytic   dissociation   the- 
ory, 213,  412 

and  freezing-point  depression,  203 

and  lowering  of  vapor  pressure,  189 


Osmotic  pressure,  measurement  of, 

176,  181 

and  nature  of  membrane,  178 
theoretical  value  of,  178 
Over  voltage,  519 
Oxidation,  definition  of,  493 
Oxidation   and   reduction    elements 
492 

Partial  pressures,  law  of,  151,  329 

of  mixed  gases,  151 

of  mixed  vapors,  158 
Particles  —  a  counting,  576 
Peptization,  270 
Periodic  law,  548 

applications  of,  552 

defects  in,  554 

and  octet  theory,  603 
Periodic  table  of  Harkins,  594 
Periodicity  of  physical  properties,  551 
Periodicity  of  radio-elements,  589 
Phase,  external,  253 

internal,  254 
Phase  rule,  332 

derivation  of,  333 
Phases,  322 

continuous  and  discontinuous,  254 
Phosphorescence,  531 
Photochemical,  action,  532 

after-effect,  542 

extinction,  534 

induction,  538 

reactions,  classification  of,  538 

reactions,  kinetics  of,  535 

sensitization,  541 

synthesis,  543 
Photochemistry,  524 
Photoelectric  cells,  544 
Photo-stationary  state,  537 
Physical    properties    of    completely 

ionized  solutions,  218 
Physical     properties     and     electron 

configuration,  608 
Physical  properties  of  gels,  255 
Physical   properties,    periodicity   of, 

551 

Poisons,  380 
Polarimeter,  114 


INDEX  OF  SUBJECTS 


623 


Polarization,  511 

capacity  of  electrodes,  513 

and  electrolysis,  511 

electromotive  force  of,  511 

theory  of,  516 
Polymorphism,  90 
Positive  column,  557 
Positive  rays,  564 

Potential  between  metal  and  solution, 
464 

measurement  of,  473 
Potential  decomposition,  511 
Potential   difference   at  junction   of 

two  solutions,  478 

Potential  difference  at  liquid  junc- 
tions, 481 

Potential,  normal  electrode,  484 
Precipitation    of    colloids    by    elec- 
trolytes, 246 

of  emulsoids,  248 
Precipitation,  mutual,  252 

of  suspensoids,  247 

and  valence  in  colloids,  248 
Preparation    of    colloidal    solutions, 

268 

Principle  of  maximum  work,  133,  298 
Promoters,  380 
Protective  colloids,  250 
Properties    of,     completely    ionized 
solutions,  217 

of  crystals,  88 

of  solids,  85 
Proto-elements,  554 

Quanta,  98 
Quantum  theory,  98 

Radiant  energy,  524 

and  catalysis,  382 

source  of,  525 
Radiations,  pure  temperature,  525 

nature  of,  572 

phosphorescence  induced  by,  572 

photographic  action  of,  572 
Radiator,  full,  529 
Radiators,  525 

Radioactive  change,  ultimate  prod- 
uct of,  582 


Radioactive  constant,  575 
Radioactive  equilibrium,  579 
Radioactivity,  569 

discovery  of,  569 
Radio-elements,    periodicity  among, 

589 

Radio-luminescence,  532 
Radium,  discovery  of,  570 

energy  evolved  by,  585 

origin  of,  580 
Radium  emanation,  574 
Rays,  a,  0,  7,  573 

Reaction,  determination  of  order  of, 
372 

influence  of  solvent  on  velocity  of, 

377 

^  isochore,  315,  439 
^  isochore,  derivation  of,  315 

isotherm,  306 

isotherm,  derivation  of,  307 

pressure,  72 

velocity  of,  360 

velocity  of,  and  temperature,  376 
Reactions,  bimolecular,  367 

at  constant  pressure,  285 

at  constant  volume,  285 

consecutive,  373 

counter,  373 

of  first  order,  365 

of  second  order,  367 

of  third  order,  370 

of  higher  orders,  371 

photochemical,     classification     of, 
538 

side,  373 

trimolecular,  370 

unimolecular,  361 

velocity  of  heterogeneous,  375 
Reduced  equation  of  state,  61 
Reduction,  definition  of,  493 
Refraction,  double,  88 

index  of,  105 

molecular,  108 

specific,  108 

Refractive  power  of  liquids,  105 
Refractometer,  Pulfrich,  106 
Residual  current,  513 
Resistance  capacity,  407 


624 


INDEX  OF  SUBJECTS 


Retrograde  solubility,  351 

Reversible  cells,  458 

cycles,  135 

processes,  134 
Rotation,  magnetic,  121 

molecular,  115 

molecular  magnetic,  122 

of  plane-polarized  light,  113 

specific,  115 

Saturated  solutions,  152 

Saturation  current,  572 

Semi-permeable  membranes,  174 

Side-reactions,  373 

Solation,  228 

Solids,  general  properties  of,  85 

heat  capacity  of,  95 
Sols,  227 
Solubility,  change  of,  170 

coefficient,  154 

product,  440 

retrograde,  351 
Solutes,  abnormal,  210 
Solution,  association  in,  210 

dissociation  in,  211 

heat  of,  282 

methods  for  colloids,  270 

molecular  weight  in,  207 

of   a   solid   in   a   non-dissociating 
solvent,  330 

pressure,  461 
Solutions,  149 

adsorption  in,  262 

buffer,  503 

classification  of,  151 

colloidal,  149 

dilute,  and  osmotic  pressure,  174 

of  electrolytes,  210 

expressions    for     composition    of, 
150 

of  gases  in  gases,  151 

of  gases  in  liquids,  152 

of  liquids  in  liquids,  155 

of  solids  in  liquids,  167 

imbibition  in,  260 

isohydric,  435 

labile,  169 

metastable,  169 


Splutions,    properties  of  completely 
ionized,  217 

saturated,  152 

solid,  170 

supersaturated,  168 

thermoneutrality  of  salt,  293 
Solvents,  ionizing  power  of,  426 
Specific  conductance,  404 
Specific  gas  constant,  22 
Specific  heat,  42,  97 

and  atomic  weight,  11 

at  constant  pressure  and  volume, 
43,97 

variation    of,    with    temperature, 

287 

Specific   heats,   and   kinetic   theory, 
49 

ratio  of,  45 

ratio  of,  determination  of,  46 
Specific  refraction,  108,  219 

of  mixtures,  113 
Specific  rotation,  115 
Spectra,  absorption,  123 
Spectrograph,  123 

mass,  565 
Standard  cells,  467 
Steam  distillation,  167 
Storage  cells,  505 
Strength  of  acids  and  bases,  432 
Sublimation,  85 
Submicrons,  228 
Supersaturated  solutions,  168 
Supersaturation,  345 
Surface  concentration,  265 

and  surface  tension,  266 
Surface  energy  of  colloids,  264 
Surface  tension,  72,  74,  219 

of  colloidal  solutions,  239 

determination  of,  75 

and  drop  weight,  79 

and  molecular  weight,  76 
Suspensions,  228 
Suspensoids,  228 

precipitation  of,  247 
Systems,  lyophile,  228 . 

lyophobe,  228 

two-component,  340 

three-component,  348 


' 


INDEX  OF  SUBJECTS 


625 


Theorem  of  Le  Chatelier,  298 
Thermal  expansion,  219 
Thermal  units,  273 
Thermochemical  equations,  274 

measurements,  275 
Thermochemistry,  273 
Thermodynamics,  130)( 

first  law  of,  43,  131 

second  law  of,  136 
Thermo-luminescence,  531 
Thermometry,  high  temperature,  529 
Thermoneutrality  of  salt  solutions, 

293 

Titration,  electrometric,  502 
Transference  numbers,  400 

electrometric  determination  of,  486 

experimental  determination  of,  400 

and  ionic  hydration,  403 

true,  403 

Transition  point,  333,  337 
Transmission  coefficient,  534 
Triads  of  Dobereiner,  547 
Tribo-luminescence,  532 
Trimolecular  reactions,  370 
Triple  point,  335 
Tyndall  phenomenon,  229 

Uitrafiltration,  230 
Ultramicroscope,  228 
Unimolecular  reactions,  361 
Units,  electrical,  392 

Valence,  14 

electrometric  determination  of,  486 

and  octet  theory,  607 
Vapor,  saturated,  68 
Vapor  density,  18,  33 

abnormal,  38,  210 

determinations  of,  18,  34,  37 

lowering  of,  and  dissociation,  215 
Vapor  pressure,  apparatus,  191 

of  binary  mixtures,  158 

of  liquids,  68 

lowering  of,  187 

lowering  and  osmotic  pressure,  189 


Vapor  pressure,  of  solutions,  meas- 
urement of,  191 

Vapor  pressures,  by  dynamic  method, 
69,  191 

by  static  method,  69,  191 
Vaporization,  heat  of,  72 
Variation    of    equilibrium    constant 
with  temperature,  330 

of  heat  of  reaction  with  tempera- 
ture, 289 

of  specific  heat  with  temperature, 

287 

Velocity,  absolute  ionic,  416 
Velocity  of  cathode  particle,  560 
Velocity  constant,  305 
Velocity  of,  gaseous  molecule,  31 

of  heterogeneous  reactions,  375 

of  imbibition,  258 

of  migration  of  suspensoids,  244 

of  reaction,  360 

of  reaction,   influence  of  the  sol- 
vent on,  377 

of  reaction  and  temperature,  376 

of  sound  in  a  gas,  47 
Vibration,  curves  of  molecular,  125 
Viscosity,  80 

of  colloidal  solutions,  239 
Voltaic  pile,  389 

Volume,  gram-molecular,  or  molar,  9 
Volumes,  Gay-Lussac's  law  of,  8,  21 

27 
Water,  dissociation  of,  421 

equilibrium  between  phases  of,  334 

ionization  constant  of,  451 

primary   decomposition  of,   520 
Water  of  crystallization,  221 
Weight,  gram-molecular,  9 

molar,  9 
Weights,  atomic,  7,  10,  14,  18 

combining,  7 

molecular,  9,  33 

X-rays,  569 

and  atomic  structure,  587 
and  crystal  structure,  91 


RES" 

from  w      ^      *  T  TBRA. 
Edue  on  the  LAST  DATE 
below. 

14  DAY  USE 

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